applying the pythagorean theorem homework 3 answers

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Pythagorean Theorem Worksheet

48 pythagorean theorem worksheet with answers [word + pdf].

The simplicity of the Pythagorean Theorem worksheet is the best thing about it. What is the Pythagorean Theorem? Formulated in the 6th Century BC by Greek Philosopher and mathematician Pythagoras of Samos, Pythagorean Theorem is a mathematic equation used for a variety of purposes. Over the years, many engineers and architects have used Pythagorean Theorem worksheet to complete their projects .

Table of Contents

• 1 Pythagorean Theorem Worksheet
• 2 Knowing Pythagoras of Samos and how he came up with the Pythagorean equation
• 3 Understanding Pythagorean Theorem
• 4 Pythagorean Theorem Word Problems Worksheet
• 5 Using Pythagorean Theorem worksheet
• 6 Conclusion

A simple equation, Pythagorean Theorem states that the square of the hypotenuse (the side opposite to the right angle triangle) is equal to the sum of the other two sides . Following is how the Pythagorean equation is written:

In the aforementioned equation, c is the length of the hypotenuse while the length of the other two sides of the triangle are represented by b and a. Though the knowledge of the Pythagorean Theorem predates the Greek Philosopher, Pythagoras is generally credited for bringing the equation to the fore. This is the reason the Pythagorean equation is named after him. Before we discuss the Pythagorean Theorem and the Pythagorean Theorem worksheet in detail, let’s take a look at who Pythagoras of Samos was and how he came up with the Pythagorean equation.

Knowing Pythagoras of Samos and how he came up with the Pythagorean equation

A 6 th century BC Greek philosopher and mathematician, Pythagoras of Samos is widely credited for bringing the Pythagorean equation to the fore. Though others used the relationship long before his time, Pythagoras is the first one who made the relationship between the lengths of the sides on a right-angled triangle public. This is why he’s regarded as the inventor of the Pythagorean equation.

Apart from being a philosopher and mathematician, Pythagoras founded the Pythagoreanism movement. Born in Croton, Italy, Pythagoras travelled to many different countries including Greece, Egypt, and India. After moving back to Croton in 530 BC, Pythagoras established some kind of school. He returned to Samos in 520 BC. It was in late 6 th Century BC that Pythagoras started to make important contributions to philosophy and math. The Pythagorean equation was one of those contributions.

Though he revealed the Pythagorean equation to the world in the late 6 th Century BC while living in Samos, many historians believe that Pythagoras first thought about the equation during his time in Egypt. In fact, according to many historians, Pythagoras learned geometry, the Phoenicians arithmetic and other branches of mathematics from the Egyptians.

Though he has made many important contributions to philosophy, Pythagoras is widely known as the founder of the Pythagorean Theorem. As previously mentioned, the Pythagorean Theorem is a mathematical equation that states that the square of the hypotenuse (the side opposite to the right angle triangle) is equal to the sum of the other two sides .

Today the aforementioned equation bears Pythagoras’s name but it’s important to know that he wasn’t the first one to use the equation. Before Pythagoras’s time, the Indians and the Babylonians utilized the Pythagorean Theorem or equation. Since they constructed the first proof of the theorem, Pythagoras and his disciples are regarded as the inventors of the equation.

Many historians say that Pythagoras worked in a very secretive manner. This is the reason little evidence is available that the Greek Philosopher/ mathematician himself worked on and proved the Pythagorean Theorem. It is important to note that the first time Pythagoras was given credit for the Theorem was five centuries after his death. This makes Pythagoras’s contribution to the Theorem even more debatable. Nonetheless, since Pythagoras is the only one connected to the Pythagorean Theorem known today, we have to give him due credit. Now that we’ve discussed who Pythagoras of Samos was and how he came up with the Pythagorean equation, it’s time to take a detailed look at the Pythagorean Theorem and the Pythagorean Theorem worksheet.

Understanding Pythagorean Theorem

According to Pythagorean Theorem, the sum of the squares on the right-angled triangle’s two smaller sides is equal to the side opposite to the right angle triangle (the square on hypotenuse). Using a Pythagorean Theorem worksheet is a good way to prove the aforementioned equation. An amazing discovery about triangles made over two thousand years ago, Pythagorean Theorem says that when a triangle has a 90° angle and squares are made on each of the triangle’s three sides, the size of the biggest square is equal to the size of the other two squares put together! A short equation, Pythagorean Theorem can be written in the following manner:

In Pythagorean Theorem, c is the triangle’s longest side while b and a make up the other two sides. The longest side of the triangle in the Pythagorean Theorem is referred to as the ‘hypotenuse’. Many people ask why Pythagorean Theorem is important. The answer to this is simple: you’ll be able to find the length of a right-angled triangle’s third side if you know the length of the other two sides. This equation works like magic and can be used to find any missing value. Following is an example that uses the Pythagorean Theorem to solve a triangle.

In this equation, the longest side of the triangle ‘c’ is missing. By finding out the sum of the squares of the two other sides, we were able to find the missing value. The most famous mathematical contribution of Pythagoras, the Pythagoras Theorem was one of the earliest documented theorems. Though Pythagoras is given most of the credit for the theorem, a major contribution to the theorem was made by his students.

When you look at a Pythagoras Theorem worksheet, you’ll notice that the theorem enables you to find the length of any right angle triangle side provided you know the length of the other two sides. Also, using the theorem, you can check whether a triangle is a right triangle. The Pythagoras Theorem is extremely useful in solving many math problems. Further, you can use it in many real life situations. This is illustrated by a Pythagoras Theorem worksheet.

Pythagorean Theorem Word Problems Worksheet

Using Pythagorean Theorem worksheet

A good way to review the Pythagoras Theorem and expand the mathematical equation is using a Pythagoras Theorem worksheet. By using the worksheet, you’ll be able to get a good understanding of geometry. Additionally, the worksheet will give you an opportunity to review the knowledge related to the different types of triangles. Finally and most importantly, you’ll be able to practice the ancient equation invented by the Greek mathematician and philosopher, Pythagoras. Before you start using the Pythagoras Theorem worksheet, just remember that ‘c’ is the hypotenuse while the shorter sides of the triangle are represented by ‘a’ and ‘b’.

A Pythagoras Theorem worksheet presents students with triangles of various orientations and asks them to identify the longest side of the triangle i.e. the hypotenuse. As you know by now, the formula used in Pythagoras Theorem is a²+b²=c². Regardless of what the worksheet asks the students to identify, the formula or equation of the theorem always remain the same. Though, the students could be presented with different challenges including solving triangles:

• Labeled in different order
• With a different set of letters
• By using vertices to name the sides

The symbols used in the Pythagoras Theorem are something students will find on their calculators. Figuring out how to use these functions is what students need to establish. There is involvement of the Babylonians and the Egyptians in the invention of the Pythagoras Theorem but the earliest known proof of the theorem was produced by the school of Pythagoras.

Many Pythagorean triples were known to the Babylonians while the Egyptians knew and used the (3, 4, 5) triple. The Chinese and Indians also played a role in the invention of the Pythagoras Theorem. The first diagrammatic proof of the theorem was produced by the Chinese while the Indians discovered many triples. In 1995, the theorem became part of the Guinness Book of Records as the most proved theorem of all time.

The triples used in the Pythagoras Theorem include (3, 4, 5), (6, 8, 10), (5, 12, 13), (8,15,17), (7,24,25), (20,21,29), (12,35,37), (9,40,41), (28,45,53), (11,60,61), (16,63,65), (33,56,65) and (48,55,73). The aforementioned triples aren’t multiples of a smaller triple and the name given to them is ‘primitive’ triples. To solve a particular problem, the Pythagoras Theorem can be arranged. For example, if you’re asked to find b which is one of the two smaller sides of the right-angled triangle, you can rearrange the theorem to b²=c²-a². By doing this, you’ll be able to easily find the missing value.

The Pythagoras Theorem has many different proofs. However, when checking your answers, following are the two things that you must always remember:

• The side opposite to the right angle or simply the hypotenuse is always the longest side of the triangle
• Though it is the longest side of the triangle, the size of the hypotenuse can never exceed the sum of the other two squares

To understand this better, take a look at a Pythagoras Theorem worksheet. Today, you can get easy access to Pythagorean Theorem worksheet with answers. Nonetheless, we’re going to try and understand the Pythagoras Theorem as much as we can.

As mentioned earlier, if you know the size of the other two sides, you will be able to find out the length of the third side of the right angle triangle. Also, after being squared, the shorter length is subtracted from the square of the hypotenuse when the hypotenuse is one of the two known lengths. As seen earlier, the lengths of each side of the triangle in the Pythagoras Theorem are whole numbers. Such triangles are known as Pythagorean triangles.

Though there are many different proofs of the Pythagoras Theorem, only three of them can be constructed by students and other people on their own. The first proof starts off as rectangle and is then divided into three triangles that individually contain a right angle. To see first proof, you can use a computer or something as straight forward as an index card cut up into right triangles.

Beginning with a rectangle, the second proof of the Pythagoras Theorem starts off by constructing rectangle CADE with BA=DA. This is followed by the construction of the <BAD’s angle bisector. Once constructed, the bisector is allowed to intersect ED at point F. This makes <BAF and <DAF congruent, BA=DA, and AF=AF. This in turn makes the triangle DAF equal to triangle BAF which means that since ADF is a right angle, ABF will also be a right angle. The third and final proof of the Pythagorean Theorem that we’re going to discuss is the proof that starts off with a right angle. In this proof, triangle ABC is right angle and its right side is angle C.

The three proofs stated above are just few of the many Pythagoras Theorem. You’ll come across these proofs when you take a look at the Pythagorean Theorem worksheet with answers. Learning and understanding the Pythagorean concept is extremely important for students and other people who’ll use this theorem in their practical life.

It is important that you understand the algebraic representation of the Pythagoras Theorem as well as the geometric concepts behind it. You can accomplish this by using proofs, manipulatives, and computer technology. By using these methods to learn Pythagorean Theorem, you’ll be able to see the connections and benefit greatly.

Formulated in the 6th Century BC by Pythagoras of Samos, Pythagoras Theorem is widely used today. If you want to practice Pythagoras Theorem then you can do that easily. Pythagoras Theorem worksheets with answers are easily available and you can use these worksheets to get a good grip of the Theorem.

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Pythagorean Theorem Worksheets

These printable worksheets have exercises on finding the leg and hypotenuse of a right triangle using the Pythagorean theorem. Pythagorean triple charts with exercises are provided here. Word problems on real time application are available. Moreover, descriptive charts on the application of the theorem in different shapes are included. These handouts are ideal for 7th grade, 8th grade, and high school students. Kick into gear with our free Pythagorean theorem worksheets!

Identifying Right Triangles

Apply Pythagorean theorem to identify whether the given triangle is a right triangle. Each printable worksheet consists of six problems.

• Download the set

Pythagorean Theorem Chart

These descriptive charts explain the Pythagorean theorem with an illustration. These pdfs emphasize the relation of the theorem derived as an equation.

Pythagorean Triple

A set of three numbers is given in each problem. Grade 7 and grade 8 students need to apply the theorem and identify whether the set of numbers forms a Pythagorean triple.

Pythagorean Triple Chart

This section comprises of Pythagorean triple sets up to 100. Besides, Pythagorean triple formulas with examples are provided in the charts.

Unknown Side of a Right Triangle

Apply Pythagorean theorem to find the unknown side of the right triangle. Round the answer to the nearest tenth.

Unknown Length: Shapes

Apply the Pythagorean theorem to find the unknown length of each shape in these printable worksheets. Round the answer to the nearest tenth.

Word Problems | Level 1

Brighten your math class with this bundle of real-life word problems based on the Pythagorean Theorem. Solve each word problem by finding the missing hypotenuse of the right triangle and rounding off the answer to the nearest tenth.

Word Problems | Level 2

Presenting word problems with clear illustrations, these pdf worksheets require high school students to plug in the known values into the equation form of the Pythagorean Theorem and figure out the unknown side of the right triangle.

Related Worksheets

» Pythagorean Inequality Theorem

» Triangle

» Triangle Inequality Theorem

» Trigonometry

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Pythagorean Theorem Exercises

Pythagorean theorem practice problems with answers.

There are eight (8) problems here about the Pythagorean Theorem for you to work on. When you do something a lot, you get better at it. Let’s get started!

Here’s the Pythagorean Theorem formula for your quick reference.

Note: drawings not to scale

Problem 1: Find the value of [latex]x[/latex] in the right triangle.

[latex]10[/latex]

Problem 2: Find the value of [latex]x[/latex] in the right triangle.

[latex]4\sqrt 7 [/latex]

Problem 3: Find the value of [latex]x[/latex] in the right triangle.

[latex]3[/latex]

Problem 4: The legs of a right triangle are [latex]5[/latex] and [latex]12[/latex]. What is the length of the hypotenuse?

[latex]13[/latex]

Problem 5: The leg of a right triangle is [latex]8[/latex] and its hypotenuse is [latex]17[/latex]. What is the measure of its other leg?

[latex]15[/latex]

Problem 6: Suppose the shorter leg of a right triangle is [latex]\sqrt 2[/latex]. The longer leg is twice the shorter leg. Find the hypotenuse.

[latex]\sqrt {10} [/latex]

Problem 7: The hypotenuse of a right triangle is [latex]4\sqrt 2[/latex]. If the longest leg is half the hypotenuse, what is the length of the shortest leg?

[latex]2\sqrt 6[/latex]

Problem 8: Find the value of [latex]x[/latex] of the right isosceles triangle.

[latex]\sqrt 6[/latex]

You may also be interested in these related math lessons or tutorials:

Pythagorean Theorem

Pythagorean Triples

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Unit 8 – The Pythagorean Theorem

The Pythagorean Theorem

LESSON/HOMEWORK

LECCIÓN/TAREA

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

SMART NOTEBOOK

The Pythagorean Theorem and Its Converse

Applying the Pythagorean Theorem

Distance in the Coordinate Plane

Understanding the Pythagorean Theorem

Unit Review

Unit 8 Review

UNIT REVIEW

REPASO DE LA UNIDAD

EDITABLE REVIEW

Unit 8 Assessment Form A

EDITABLE ASSESSMENT

Unit 8 Assessment Form B

Unit 8 Exit Tickets

Unit 8 Mid-Unit Quiz – Form A

Unit 8 Mid-Unit Quiz – Form B

U08.AO.01 – Practice with the Pythagorean Theorem and Its Converse

EDITABLE RESOURCE

U08.AO.02 – Practice Finding Distance in the Coordinate Plane

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Pythagorean Theorem Worksheets

Related Topics: More Math Worksheets More Printable Math Worksheets 7th Grade Math

There are five sets of triangle worksheets:

• Triangle Sum Theorem
• Exterior Angle Theorem
• Area of Triangle

Area of Triangle using Sine

• Pythagorean Theorem

Examples, solutions, videos, and worksheets to help grade 7 students learn how to use the Pythagorean Theorem.

How to use the Pythagorean Theorem?

There are five sets of Pythagorean Theorem worksheets:

• Find the length of the hypotenuse
• Find the length of a side
• Find the length in two steps
• Check for right triangle
• Pythagorean Theorem Word Problems

The Pythagorean Theorem also called the Pythagoras’ Theorem is a fundamental principle in geometry that relates the sides of a right triangle.

It states that in a right triangle (a triangle with one 90-degree angle), the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Mathematically, the Pythagorean Theorem can be expressed as: c² = a² + b²

• c represents the length of the hypotenuse.
• a and b represent the lengths of the other two sides.

The Pythagorean Theorem is widely used to calculate the length of any one side of a right triangle when the lengths of the other two sides are known. It is an essential tool in geometry and has applications in various fields, including trigonometry, engineering, physics, and architecture.

The Pythagorean Theorem can be used in various ways, including:

• Finding length of any one side of a right triangle when the lengths of the other two sides are known.
• Checking if a Triangle is Right: You can use the theorem to check if a triangle is a right triangle by verifying if the relationship a² + b² = c² holds.
• Solving Problems Involving Right Triangles: The theorem is used to solve problems involving right triangles, such as finding distances, heights, or angles in various real-world situations.

For example, if you have a right triangle with one side (a) measuring 3 units and the other side (b) measuring 4 units, you can use the Pythagorean Theorem to find the length of the hypotenuse (c):

c² = a² + b² c² = 3² + 4² c² = 9 + 16 c² = 25

To find c, take the square root of both sides:

c = √25 c = 5

So, the length of the hypotenuse (c) is 5 units. This illustrates how the Pythagorean Theorem can be used to solve for side lengths in a right triangle.

Click on the following worksheet to get a printable pdf document. Scroll down the page for more Pythagorean Theorem Worksheets .

More Pythagorean Theorem Worksheets

Printable (Answers on the second page.) Pythagorean Theorem Worksheet #1 (find hypotenuse) Pythagorean Theorem Worksheet #2 (find a side) Pythagorean Theorem Worksheet #3 (2-step) Pythagorean Theorem Worksheet #4 (check for right triangle) Pythagorean Theorem Worksheet #5 (word problems)

Online Pythagorean Theorem Find the missing side Test for right triangle Pythagorean Theorem Word Problems Converse Pythagorean Theorem, Types of Triangles

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Unit 9: Pythagorean theorem

About this unit.

The Pythagorean theorem describes a special relationship between the sides of a right triangle. Even the ancients knew of this relationship. In this topic, we’ll figure out how to use the Pythagorean theorem and prove why it works.

Pythagorean theorem

• Intro to the Pythagorean theorem (Opens a modal)
• Pythagorean theorem example (Opens a modal)
• Pythagorean theorem intro problems (Opens a modal)
• Use Pythagorean theorem to find right triangle side lengths 7 questions Practice
• Use Pythagorean theorem to find isosceles triangle side lengths 7 questions Practice
• Right triangle side lengths 4 questions Practice
• Use area of squares to visualize Pythagorean theorem 4 questions Practice

Pythagorean theorem application

• Pythagorean theorem word problem: carpet (Opens a modal)
• Pythagorean theorem word problem: fishing boat (Opens a modal)
• Pythagorean theorem in 3D (Opens a modal)
• Use Pythagorean theorem to find perimeter 4 questions Practice
• Pythagorean theorem word problems 4 questions Practice
• Pythagorean theorem in 3D 4 questions Practice
• Pythagorean theorem challenge 4 questions Practice

Pythagorean theorem and distance between points

• Distance formula (Opens a modal)
• Distance formula review (Opens a modal)
• Distance between two points 7 questions Practice

Pythagorean theorem proofs

• Garfield's proof of the Pythagorean theorem (Opens a modal)
• Bhaskara's proof of the Pythagorean theorem (Opens a modal)
• Pythagorean theorem proof using similarity (Opens a modal)
• Another Pythagorean theorem proof (Opens a modal)

20 Pythagorean Theorem Worksheet with Answers (Word & PDF)

If you are familiar with the Pythagorean Theorem, then you know that it is one of the simplest and most useful tools in geometry. The theorem was formulated by Pythagoras of Samos, a Greek mathematician, and philosopher, in the 6th century. Today, it has many applications in different areas of study, from simple elementary and high school geometry to complex engineering and architecture calculations. Below, we take a closer look at the theorem and how you can master it through a worksheet.

What Is a Pythagorean Theorem Worksheet?

Simply put, the Pythagorean Theorem states that the square of the longest side of a right angle (the hypotenuse) is equal to the sum of the squares of the other two sides. A Pythagorean Theorem Worksheet is a workbook containing different triangle orientations requiring a student to fill in the missing values using the Pythagorean Theorem. It is a great study tool for geometry students.

Pythagorean Theorem Worksheet Templates & Examples

Understanding How Pythagoras of Samos Came Up with the Pythagorean Equation

Pythagoras of Samos was a Greek mathematician and philosopher of the 6th century who is widely credited with bringing the Pythagorean Theorem to the attention of the public. While there are other people who used the Pythagorean equation earlier than Pythagoras – most notably the Egyptians – he was the first person to make the connection between the lengths of a right angle triangle’s sides public. This explains why he is widely recognized as the originator and inventor of the equation and why it is named after him – the Pythagorean Equation.

Pythagoras of Samos shared information about the Pythagorean equation with the world in the late 6th century when he resided in Samos. However, many historians maintain that he first made the discovery during his visit to Egypt, where historians believe he learned several branches of mathematics, including Phoenicians arithmetic and geometry.

According to Historians, Pythagoras was very secretive about his work. There is very limited evidence that he even worked on the Theorem or proved it, and credit is often given to his students. Additionally, Pythagoras of Samos was only credited with proving the Theorem over 5 centuries after his demise – the fact that fuels a lot of debate on whether he was behind the theorem at all.

That said, however, Pythagoras of Samos is the most popular figure connected to the Pythagorean Theorem as it is known today. Therefore, it is only fair to give him credit for this amazing equation that has grown to have numerous real-life applications.

Understanding Pythagorean Theorem

As mentioned, the Pythagorean Theorem states that, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. The theorem basically says that if you make squares on each side of a triangle with a 90° angle, the two smaller squares put together will be the same size as the largest square.

The Pythagorean Theorem can be represented mathematically as follows:

a² + b² = c²

Here, c represents the length of the hypotenuse (the longest side), while b and a are the lengths of the other two sides. From the equation, you can easily find the value of one side if you have the values of the other two. Here is an example to demonstrate:

Q. Consider a right-angled triangle with a base of 3cm and a height of 4cm. What is the hypotenuse of the triangle?

A: Using the Pythagorean Theorem to solve, we can assign the base ‘a’ and the height ‘b.’ Substituting in the equation, we have:

a² + b² = c² 3² + 4² = c² 9 + 16 = c² 25 = c² c = √25 c = 5

The hypotenuse of the triangle is 5 cm long.

As you can see, the hypotenuse was missing from the equation, and we used the relation to find its value. Such questions can be framed differently so that you either have to calculate the value of the base or height. All you have to do is use the Pythagorean Theorem to substitute the missing value. Additionally, you can use the equation to determine whether a triangle is a right triangle.

How to Use Pythagorean Theorem Worksheet

The Pythagorean Theorem Worksheet is an excellent way to expand your understanding of the Pythagorean Theorem. As you use the worksheet, it is important to remember that ‘a’ and ‘b’ represent the shorter lengths on the triangle while c represents the hypotenuse, which is the longest side.

Pythagoras Theorem worksheets present you with different triangle orientations from which you must determine the value of the missing side. This could be the base, height, or hypotenuse. Regardless of the value you are asked to find; you can use the formula (a²+b²=c²) to solve for the answer. Remember, you can rearrange the equation, but it never changes. Several variations you could use include:

a² + b² = c² (standard) a² = c² – b² (finding the value of a) b² = c² – a² (finding the value of b)

Because the worksheet is designed to help familiarize you with the Pythagorean Theorem, the practice questions can be arranged differently to create a challenge. Some examples include:

• Questions using a different set of letters, e.g. (q, r, s) or (x, y, z)
• A question where the triangle sides are labeled in a different order.
• Questions where the sides are named as vertices.

You can also use the worksheet to prove the Pythagorean Theorem and determine whether a triangle is right-angled. As you check your answers, you should always remember:

• The hypotenuse is the longest side of the triangle and the side opposite to the 90° angle.
• While it is the longest side of the triangle, its size can never surpass the sum of the squares of the other 2 sides.

Pythagoras of Samos formulated the Pythagorean Theorem in the 6th century and, since then, it has been applied to several areas of Mathematics and Geometry. Students can use a Pythagorean Theorem Worksheet to understand the theorem and apply it to solving different questions. Depending on the student’s needs, some of these worksheets have answers, but you can set these pages aside and use them to confirm your answers.

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Pythagorean Theorem Worksheets

When a triangle has a right side (measuring ninety degrees) and you extend each side with a square, the biggest square has the same exact are as the sum of the squares of the other two sides. If we labelled the longest side c and the other two sides (legs) a and b, we could sum up that phenomena with the equation a 2 + b 2 = c 2 . We term the longest side, that opposite the right angle, the hypotenuse. As you gather by the algebra, if you know the measures of any two sides of a right triangle, you can use this theorem to determine the third side. Pythagorean Theorem lends itself to all form of building and construction because it allows you to easily determine the length of any material that we using as long as you have two materials that meet at a right angle. Sailors use this all the time to find the shortest route between two points. You will find many applications in the field of surveying when cartographers and engineers are constructing maps that indicate steepness and slopes of areas.

Aligned Standard: Grade 8 Geometry - 8.G.B.8

• Applying the Pythagorean Theorem Step-by-Step Lesson - The side that is missing is defined as the hypotenuse. We can tell this because it is directly opposite the right angle.
• In Action Guided Lesson - Example: Given a right triangle with a hypotenuse of 27 and a leg of 17, what is the measure of the missing leg?
• Guided Lesson Explanation - If this is a right triangle it will work out when the Pythagorean Theorem is applied to it. This means it will flow in the formula: leg 2 + leg 2 = hypotenuse 2
• Are They Right Triangles? - Determine the classification of these shapes.
• Skills Review - We explore every way this topic can be looked at.
• Answer Keys - These are for all the unlocked materials above.

Review Sheets

The coordinate graph makes it much more understandable than just labeling sides.

• Review 1 - We will look this from the diagram perspective and in word form.
• Review 2 - Another go at this series of skills.

Homework Sheets

• Homework 1 - We will see if these shapes will work for us.
• Homework 2 - Another approach at the first version.
• Homework 3 - All of these shapes are much more narrow.
• Homework 4 - You will find decimal lengths to work with.

Practice Worksheets

We start to find random distances between points.

• Practice 1 - You will work with some really scaled down sides.
• Practice 2 - We orientate these shapes from all directions.
• Practice 3 - This is a basic look at this skill.
• Practice 4 - More decimal lengths to work with.

Math Skill Quizzes

Here are more like problems to help you master this topic.

• Quiz 1 - Use the Pythagorean theorem to find the length of the missing side of the right triangle.
• Quiz 2 - Apply this formula to see if you can determine that missing side.
• Quiz 3 - Time break out the decimal form one more time.

How is the Pythagorean Theorem Used Every in Life?

The history behind how this theorem was conceived is almost fiction-like. Pythagoras was a Greek mathematician, among other things. He was born around 500 B.C. He founded a group called the Brotherhood of Pythagoreans which was devoted towards the study of mathematics. The theorem is credited to Pythagoras, but historians do strongly suggest that it was the effort of this group.

More than 2,500 years later this math still holds up and we use this theorem in our daily lives without even realizing it. It goes without saying that most forms of architecture and construction projects heavily rely on this to form straight lines in their buildings. It also is the foundation of how the angle of roofing is pitched on almost all structures. All the beams and materials used are dependent on this calculation. Something as simple as determining how to optimal position ladders against a wall for safety are calculated in this manner as well.

The Brotherhood of Pythagoreans probably never thought that large water vessels would depend on their math to circumnavigate the Earth's oceans. From very short distances to long, understanding the best path is often a product of this theorem. It also helps ships Captains determine how to interest or disrupt enemies in battle. Watch just about any war movie that involves a ship or a submarine and you will see mathematical compasses and rulers putting this to work.

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Pythagoras Theorem Questions (with Answers)

Pythagoras theorem.

In a right triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.

• Length of the hypotenuse is c
• The hypotenuse is the longest side
• Lengths of the other sides are a, b

Right Triangle Questions – using the theorem

The Theorem helps us in:

• Finding Sides: If two sides are known, we can find the third side.
• Determining if a triangle is right-angled: If the sides of a triangle are known and satisfy the Pythagoras Formula, it is a right-angled triangle.

There is a proof of this theorem by a US president. Its simplicity makes it is easy enough for the grade 8 kids to understand.

Finding the missing sides (side lengths) of a Right Triangle

The theorem gives a relation among the three sides of a right-angled triangle. We can find one side if we know the other two sides. How?

Example: We are given (see figure) below the two sides of the right triangle. Find the third side.

Given: a = 3, c = 5

Which side is the hypotenuse?

✩ Always identify the hypotenuse first

Unknown side = BC = b ?

Putting the values in the Pythagoras Formula: a 2 + b 2 = c 2

3 2 + b 2 = 5 2

9 + b 2 = 25

b 2 = 25 − 9 = 16 = 4 2

Finding the Hypotenuse of a Triangle

Using the Pythagoras formula, finding hypotenuse is no different from any other side.

Example: Sides of a right triangle are 20 cm and 21 cm, find its hypotenuse.

Pythagoras Formula: a 2 + b 2 = c 2

• c is the length of the hypotenuse
• a, b are the lengths of the other two sides (you can assume any length as a or b ).

Let AB = a = 20, BC = b = 21

Putting values in the formula:

20 2 + 21 2 = c 2

400 + 441 = c 2

Finding Right Triangle

Given the sides, we can determine if a triangle is right-angled by applying the Pythagoras Formula. How?

• Assume the longest side to be hypotenuse Length = c . Find its square ( = c 2 )
• Find the sum of squares of the other two sides ( = a 2 + b 2 )
• If a 2 + b 2 ≠ c 2 it is a not right triangle
• If a 2 + b 2 = c 2 it is a right triangle

Example: A triangle has sides 8 cm, 11 cm, and 15 cm. Determine if it is a right triangle

Longest side = 15 cm. Let us assume it to be hypotenuse = c (as we know that it is always the longest)

c 2 = 15 2 = 225

Other sides a = 8 cm b = 11 cm. (You can assume any side length to be a or b ).

a 2 + b 2 = 8 2 + 11 2 = 64 + 121 = 185

a 2 + b 2 ≠ c 2

So this is not a right-angled triangle.

Example: The sides of a triangle are 8 cm, 17 cm, and 15 cm. Find if it is a right triangle.

Longest side = 17 cm. Let us assume it to be hypotenuse = c

c 2 = 17 2 = 289

Other sides a = 8 cm, b = 15 cm.

a 2 + b 2 = 8 2 + 15 2 = 64 + 225 = 289

So a 2 + b 2 = c 2

It is a right-angled triangle.

Pythagoras Questions Types

You will encounter the following types of questions related to this theorem:

• Find a side, given two sides These questions are the direct application of the theorem (formula) and are easiest to solve.
• Express the relation between the two sides in an equation
• Substitute one side by the other using the first equation in the Pythagoras Formula
• Given the Perimeter and one side, find other sides – Perimeter is the sum of the three sides. Since one side is known, we subtract it from the perimeter to get a relationship between the other two sides.
• Given Area and one side find other sides – Area = 2 1 ​ × ( ba se × a lt i t u d e ) . Base and altitude can be the sides with the right angle OR the hypotenuse and the altitude.

Pythagoras Questions

The questions chosen have minimal use of other concepts, yet, some of these are hard Pythagoras questions (See Ques 4 and Ques 10 ).

1 Question

ABC is a right triangle. AC is its hypotenuse. Length of side AB is 2√5 . Side BC is twice of side AB. Find the length of AC.

Can you express BC in terms of AB and apply the Pythagoras Theorem?

Ans.   AC = 10

2 Question

The hypotenuse of a right triangle is 6 cm . Its area is 9 cm 2 . Find its sides.

Can you form two equations – one using area and the other using the Pythagoras formula?

Ans.   Each side is 3√2 cm.

3 Question

One side of a right triangle is 4 10 ​ cm. Find the length of its other side if the hypotenuse is 13 cm.

Can you directly apply the Pythagoras Theorem?

Ans.   3 cm

4 Question

In a right triangle ABC, length of the medians to the sides AB and BC are 2 61 ​ and 601 ​ respectively. Find the length of its hypotenuse.

CE = 2 61 ​

Let AE = EB = x

BD = DC = y

Can you use the Pythagorean Theorem to form an equation between x, y, and AD?

Can you form another equation involving x, y, and EC?

Can you use this result to find A C 2 ? Use the triangle ABC.

Ans.   AC = 26

5 Question

In a right triangle, the longest side is 8 cm. One of the remaining sides is 4√3 cm long. Find the length of the other side.

Can you apply the Pythagoras Theorem directly?

Ans.   4 cm

6 Question

The first side of a right triangle is shorter than the second side by 1 cm. It is longer than the third side by 31 cm. Find the sides of the triangle.

Can you form equations between the first side and the other two sides? Which side is the hypotenuse?

Ans.   9 cm, 40 cm, 41 cm

7 Question

The perimeter of a right triangle is equal to 30 cm. The length of one of its sides is 10 cm. Find its hypotenuse.

Can you find the relation between the unknown side and the perimeter in terms of the hypotenuse?

Ans.   12.5 cm

8 Question

The sides of a triangle are 5 cm, 9 cm, and 12 cm. Is it a right-angled triangle?

Can you identify the possible hypotenuse? Also, test if the sides satisfy the Pythagoras Formula.

Ans.   Not a right triangle.

9 Question

In a right triangle, two sides are equal. The longest side is 7√2 cm, find the remaining sides.

Ans.   7 cm

10 Question

In the following right triangle altitude BD = 9 10 ​ cm and DC = 27 10 ​ cm. Find the sides of the triangle.

Can you apply the Pythagoras Theorem to the triangle BCD?

Can you form an equation between a and x using the triangle ABD?

Can you find the value of x using above equations and triangle ABC?

Ans.   AB = 30cm, BC = 90cm, A C = 30 10 ​ c m

Answers to Pythagoras Questions

1 answer.

Let AB = a , BC = b and AC = c .

AB = a = 2√5

BC is twice of AB, b = 2a = 4√5

AC = Hypotenuse = c

Applying the Pythagoras Theorem a 2 + b 2 = c 2 :

(2√5) 2 + (4√5) 2 = c 2

4(5) + 16(5) = c 2

c 2 = 20 + 80 = 100

2 Answer

Let AB = a , BC = b

In triangle ABC, base = b and altitude = a

Area of Triangle = 2 1 ​ × ( ba se × a lt i t u d e ) . So:

2 1 ​ × ( ab ) = 9

ab = 18 (Equation 1)

Using the Pythagoras Formula:

a 2 + b 2 = 6 2 = 36

We add and subtract 2ab to complete the square :

a 2 + b 2 − 2ab + 2ab = 36

(a − b) 2 + 2ab = 36

(a − b) 2 + 36 = 36 Using ab = 18 from Equation 1

(a − b) 2 = 0

Substituting a by b in Equation 1:

Side AB = BC = 3√2 cm

3 Answer

Let the length of sides be a, b and c , such that:

a = 4 10 ​ cm

b = unknown

From the Pythagoras formula a 2 + b 2 = c 2 , we get:

( 4 10 ​ ) 2 + b 2 = 1 3 2

(16 × 10) + b 2 = 169

160 + b 2 = 169

b 2 = 169 − 160 = 9

4 Answer

Using triangle ABD:

AB 2 + BD 2 = AD 2

( 2 x ) 2 + y 2 = ( 601 ​ ) 2

4 x 2 + y 2 = 601 (Equation 1)

Using triangle EBC :

EB 2 + BC 2 = EC 2

x 2 + ( 2 y ) 2 = ( 2 61 ​ ) 2

x 2 + 4 y 2 = ( 2 61 ​ ) 2

x 2 + 4 y 2 = 244 (Equation 2)

Adding Equation 1 and 2:

4x 2 + y 2 + x 2 + 4y 2 = 601 + 244

5x 2 + 5y 2 = 845

x 2 + y 2 = 169 (Equation 3)

Let us solve for the hypotenuse using the triangle ABC:

AB 2 + BC 2 = AC 2

(2x) 2 + (2y) 2 = AC 2

4x 2 + 4y 2 = AC 2

4(x 2 + y 2 ) = AC 2

Substituting the value of x 2 + y 2 from Equation 3:

4(169) = AC 2

A C = 4 ( 169 ) ​

AC = 2 × 13 = 26

5 Answer

Let the lengths of sides be a, b and c (hypotenuse).

Hypotenuse is the longest side. So c = 8 .

Let b = 4√3 .

From the Pythagoras Theorem:

a 2 + b 2 = c 2

a 2 + (4√3) 2 = 8 2

a 2 + 16(3) = 64

a 2 + 48 = 64

The third side is 4 cm.

6 Answer

The second side is the longest. It is the hypotenuse. Let its length be c .

Let the length of first side be b and third side a .

Applying the Pythagoras formula:

(b − 31) 2 + b 2 = (b + 1) 2

b 2 − 62b + 31 2 + b 2 = b 2 + 2b + 1

b 2 − 62b + 961 = 2b + 1

b 2 − 64b + 960 = 0

b 2 − 24b − 40b + 960 = 0

b(b − 24) − 40(b − 24) = 0

(b − 40)(b − 24) = 0

b = 40 Or b = 24

For b = 24 , we get a = 24 − 31 = − 7 . Length of a side cannot be negative, so we reject b = 24 .

For b = 40 , we get a = 40 − 31 = 9 and c = 40 + 1 = 41

The sides of triangle are 9 cm, 40 cm and 41 cm.

7 Answer

Side BC = b = 10 cm

Perimeter = Sum of the sides

= a + b + c = 30 (Given)

a + 10 + c = 30

a = 20 − c ( Equation 1 )

Applying the Pythagoras Theorem to find the hypotenuse:

Using Equation 1 to substitute the value of a

(20 − c) 2 + (10) 2 = c 2

400 − 40c + c 2 + 100 = c 2

500 − 40c = 0

The length of hypotenuse = 12.5 cm

8 Answer

Longest side = 12 cm. Let us assume it to be the hypotenuse = c

So c 2 = 12 2 = 144

The Pythagoras Formula: a 2 + b 2 = c 2

We can assume any side to be a or b.

Let a = 5 cm, b = 9 cm.

a 2 + b 2 = 5 2 + 9 2 = 25 + 81 = 106

So a 2 + b 2 ≠ c 2

This is a not a right angled triangle.

9 Answer

Let the length of the sides be a, b , and c (hypotenuse).

In a right triangle hypotenuse is the longest side. So c = 7√2

Other sides are equal. So a = b .

Applying the Pythagoras theorem:

b 2 + b 2 = (7√2) 2

2b 2 = 49(2)

Each side is 7 cm.

10 Answer

Let AB = a, BC = b, AC = c and AD = x

Given B D = 9 10 ​ D C = 27 10 ​

Applying Pythagoras Theorem to triangle BCD:

BD 2 + DC 2 = BC 2

( 9 10 ​ ) 2 + ( 27 10 ​ ) 2 = B C 2

(9 2 × 10) + (27 2 × 10) = b 2

810 + 7290 = b 2

Applying Pythagoras Theorem to triangle ABD:

BD 2 + AD 2 = AB 2

a 2 = ( 9 10 ​ ) 2 + x 2 (Equation 1)

From the figure:

AC = AD + BD

c = x + 27 10 ​ (Equation 2)

Applying Pythagoras Theorem to triangle ABC:

Using b = 90 and value of a 2 from Equation 1 and c from Equation 2:

( 9 10 ​ ) 2 + x 2 + 9 0 2 = ( x + 27 10 ​ ) 2

9 2 ( 10 ) + x 2 + 9 0 2 = x 2 + 54 10 ​ x + 2 7 2 ( 10 )

810 + 8100 = 54 10 ​ x + 7290

54 10 ​ x = 8910 − 7290 = 1620

x = 10 ​ 30 ​

Putting value of x in Equation 1:

a 2 = ( 9 10 ​ ) 2 + ( 3 10 ​ ) 2

a 2 = 810 + 90 = 900

Using value of a and b in Pythagoras Formula for triangle ABC:

30 2 + 90 2 = c 2

900 + 8100 = c 2

c = 30 10 ​

AB = 30cm, BC = 90cm, A C = 30 10 ​ c m

Difficult Pythagoras Questions (Year 10, Guided Answers) ➤

James Garfield Pythagorean Theorem (Illustration & Proof) ➤

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Maneuvering the Middle

Student-Centered Math Lessons

Read Before Teaching Pythagorean Theorem

Let’s talk about my favorite theorem, Pythagorean’s! Though it ranked 4 out of 100 in mathematicians, Paul and Jack Abad’s, list of 100 Greatest Theorems , I know it ranks number 1 in most middle school math teachers’ hearts.

Vertical Alignment

Before I jump into any topic, I like to see what students already know. Our units are complete with vertical planning, so I don’t have to search far to see that students have used equations to solve for unknown angles. Now we just need to make the jump to side lengths!

Plethora of Hook Opportunities

There are so many compelling ways to introduce Pythagorean Theorem to your students. In fact, the more context and real-world examples you can provide about the math they are learning, the more investment your students will have.

This Who Wants to Be a Millionaire video clip is kind of silly, but is a short and easy way to exclaim, “YOU MIGHT BE ASKED THIS ON A GAME SHOW FOR MONEY” or more likely, be asked on the street by someone making a video for TikTok. 

Day Mayer’s Three Act – The Taco Cart was recommended by many teachers in a Middle School Math Facebook group. I would show the first video and present it as, “By the end of this unit, you will be able to solve this problem.” This Desmos Adaptation is a great digital version.

Go Hands-On

Pythagorean Theorem is incredibly visual, but it is also incredibly kinetic. Let’s encourage students to manipulate and play to develop an understanding of how and why the Pythagorean Theorem works . You can do this in a variety of ways:

• Cheez Its or Starbursts

• Graph Paper – This PBS website gives you an outline on how to do this.
• Pythagorean Theorem Intro Activity – Another way students can “see” the theorem in action.

Know the Parts

One of the most common misconceptions students will face is identifying the legs and the hypotenuse on a triangle especially when the triangle is orientated in a different way.

Provide ample opportunities for students to label and annotate the right triangle before moving onto any calculations. Here is a non-exhaustive list of all the ways to help students labeling parts correctly:

• hypotenuse^2 = leg ^2 + leg ^2

• Hypotenuse is the longest word and therefore the longest side
• Have students draw an arrow directly across the right angle (like I did in pink above) to the hypotenuse and label it first thing as hypotenuse = c first! This reinforces the idea that students should double check that it is a right angle before assuming it is a right angle.
• If a student feels hung up on the orientation of a right triangle (because it isn’t “right”), encourage students to turn their paper until the right triangle looks “right” to them.

Practice, Practice, Practice

There is a hierarchy of math practice. Students need to build up fluency, so they are able to tackle the more complex problem types. If they can’t square a number, take a square root of a number, or solve an equation for a variable, then they will struggle to solve application problems involving Pythagorean Theorem.

Pythagorean Theorem Maze Practice – This practice is scaffolded. One maze uses solely whole numbers, and then students can move on to problems involving rational numbers. 

Pythagorean Theorem He Said, She Said Activity – Error analysis is beneficial for procedural and conceptual understanding. It allows students to think behind the HOW and into the WHY. These types of problems are great for partner work and encourage mathematical discourse. 

Pythagorean Theorem Performance Task – If you are looking for an alternative to a unit test, may I recommend this math task? It provides multiple opportunities to show mastery of the Pythagorean Theorem, and it will encourage students to use multiple strategies of problem solving. What is the task? Students must design the layout of various attractions at a Harvest Festival. If you haven’t assigned a performance task in your classroom, be sure to check out these performance tasks tips and tricks .

Other Pythagorean Theorem Tips

• Does this side length make sense? Reasonableness is always something to be challenging students to consider. This develops their number sense (which is important when 8th graders are using calculators). When students are solving equations that are multi-step, I find that students can get so excited they leave off the last step. In the case of the Pythagorean Theorem, it is taking the final square root. I like to faux forget the last step and box my answer and wait for students to call me out! Then I would ask, “Does this answer make sense?” If I have side lengths of 12 and 9 inches, would the last side make sense at 225 inches?
• Posting an anchor chart of perfect squares can be a great way to reinforce students memorizing perfect squares. If students quickly know these math facts it will serve them well in Algebra 1, so this is a great time to work on that skill. Our 8th Grade Math and Algebra 1 Fast Pass have a list of perfect squares, so download the freebie and give it to your students to reference.

What tips do you have for teaching Pythagorean Theorem?

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January 19, 2024 at 10:28 am

Thanks so much for sharing your ideas, resources, and knowledge with us! I love how you present ideas to us in multiple modalities, such as hands-on ideas along with technology to incorporate. The vertical alignment is a valuable tool for planning. Thanks again!

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Into Math Grade 8 Module 11 Lesson 3 Answer Key Apply the Pythagorean Theorem

We included  HMH Into Math Grade 8 Answer Key PDF   Module 11 Lesson 3 Apply the Pythagorean Theorem to make students experts in learning maths.

HMH Into Math Grade 8 Module 11 Lesson 3 Answer Key Apply the Pythagorean Theorem

I Can apply the Pythagorean Theorem to solve real-life problems involving the legs and hypotenuse of a right triangle, including problems in three dimensions.

Step It Out

Explaantion: Given the red team and blue team are playing capture the Flag. Each team places their flag at opposite corners of the field. The red team sends two players, Alfredo and Angelina, from the red flag to capture the blue flag. Alfredo must follow the white arrows to the flag so far does he runs is 150 m + 85 m = 235 meters, Alfredo runs 235 meters.

Explanation: Given Angelina must follow the black arrow to the flag. Using the Pythagorean Theorem to determine how far she runs. a 2  + b 2  = c 2 , 150 2 + 85 2  = c 2 , 22,500 + 7,225 = c 2 , c 2 = 29,725, So c is equal to sqaure root of 29,725 we get c = 172.4 round to the nearest tenth is 172. Therefore Angelina runs 172 meters.

Explanation: Farther does Alfredo run than Angelina as Alfredo runs 235 meters and Angelina runs 172 meters is 235 – 172 = 63 meters.

Turn and Talk Will the direct route always be shorter than a route between two points with a right angle? Why or why not? Answer: No,

Explanation: A direct route isn’t always the shortest distance between two points with a right angle, The shortest distance between two points with a right angle depends on the geometry of the object/surface in question.

The Pythagorean Theorem can be used to find lengths inside a three-dimensional object by finding right triangle relationships inside the object.

Explanation: As given the length of \(\overline{D C}\) is 30 inches.

Explanation: To know the length of \(\overline{A C}\). Looking at the bottom of the box. \(\overline{A C}\) is the hypotenuse of another right triangle, ∆ABC. Using the Pythagorean Theorem to find the length of \(\overline{A C}\) as a 2  + b 2  = c 2 , 12 2 + 12 2 = ac 2 , 144 + 144 = \(\overline{A C}\) 2 , \(\overline{A C}\) 2 = 288, So ac is equal to sqaure root of 288, we get \(\overline{A C}\) = 16.97, rounding to the nearest tenth we get \(\overline{A C}\) = 17.

Explanation: As we have \(\overline{A C}\) = 17 inches and the length of \(\overline{D C}\) is 30 inches, So the length of \(\overline{A D }\) is 17 2 + 30 2 = ad 2 , 289 + 900 = \(\overline{A D}\) 2 , \(\overline{A D}\) 2 = 1,189, So ad is equal to sqaure root of 1,189, we get \(\overline{A D}\) = 34.48, rounding to the nearest tenth we get \(\overline{A D}\) = 34 inches.

D. About how long is the longest rod that could fit in the box? ___________ inches Answer: About 34 inches,

Explanation: As we have the longest distance in the box is the diagonal AD between opposite corners. It forms the hypotenuse of ∆ACD. So we have \(\overline{A D}\) = 34 inches, therefore 34 inches is the longest rod that could fit in the box.

E. Can the curtain rod fit in the box? Why or why not? Answer: No,

Explanation: As Jada wants to ship a 3-foot curtain rod to a customer, 3 foot, as 1 foot is equal to 12 inches so 3 foot means 3 X 12 = 36 inches, As the longest rod that could fit in the box is only 34 inches and we the curtain rod is 36 inches which is greater, So the curtain rod will not fit in the box.

Explanation: Given Cara measured the radius and outside length of an ice cream cone. Identified the right triangle in the cone, then used Cara’s measurements to find the height of the cone let it be b so 2.5 2 + b 2 = 11.7 2 , 6.25 + b 2 = 136.89 2 , b 2 = 136.89 – 6.25 = 130.64, So b is equal to sqaure root of 130.64, we get b = 11.42 rounding to the nearest hundredth is 11 cm. Therefore the height of the cone is about 11 centimeters.

Turn and Talk Cara has another ice cream cone. The radius and the height of the second cone is known. Explain how to use these dimensions to find the outside length of the cone. Answer: Outside length of the cone is square root of square of radius plus square of the height of the second cone,

Explanation: Given Cara has another ice cream cone. The radius and the height of the second cone is known. Using these dimensions to find the outside length of the cone is sqaure root of square of radius plus square of the height of the second cone. Let the outside length of the cone be c and radius be a and the height of the second cone is b, So c = sqaure root of a 2 + b 2 .

Check Understanding

Explanation: Given computer monitors are measured diagonally from corner to corner. If the rectangular screen of a 40-inch monitor is 35 inches wide, So let the height of the monitor is h, so applying Pythagorean theorem we have 35 2 + h 2 = 40 2 , 1,225 + h 2 = 1,600, h 2 = 1,600 – 1,225 = 375, So h is equal to sqaure root of 375, we get h = 19.36 rounding to the nearest tenth is 19 inches.

Explanation: Given a 20-inch rod fits perfectly in a box 15 inches tall. So the measurement of x in the diagram applying Pythagorean theorem we have 15 2 + x 2 = 20 2 , 225 + x 2 = 400, x 2 = 400 – 225 = 175, So x is equal to sqaure root of 175, we get x = 13.22 rounding to the nearest tenth is 13 inches.

On Your Own

Explanation: The length of the sidewalk from the library to the school is78 m  + 13 m = 91 m and the length she can travel across the grass directly is let it be x, applying Pythagorean theorem we have 13 2 + 78 2 = x 2 , 169 + 6,084 = x 2 , x 2 = 6,253 , So x is equal to sqaure root of 6,253, we get x = 79.07 rounding to the nearest tenth is 79 m. as 91 m > 79 m so the longest route is the sidewalk from the library to the school.

B. What is the difference between the lengths of the two routes? Answer: 12 m,

Explanation: As sidewalk from the library to the school distance is 91 m and the length she can travel across the grass directly is 79 m, the difference between the lengths of the two routes is 91 m – 79 m = 12 m.

Question 4. Adam’s closet measures 36 inches by 24 inches by 96 inches. What is the longest distance between corners inside his closet? Round to the nearest tenth. Answer: The longest distance between corners inside his closet is 105 inches,

Explanation: Given Adam’s closet measures 36 inches by 24 inches by 96 inches. So the longest distance between corners inside his closet is square root of length 2 + width 2 + height 2 = sqaure root of 36 2 + 24 2 + 96 2 = sqaure root of 1,296 + 576 + 9,216 = sqaure root of 11,088 = 105.29 rounding to the nearest tenth is 105 inches.

Explanation: Given the top of a building is a cone 10 meters tall with a radius of 4 meters at the base. So the measurement of x in the diagram is applying Pythagorean theorem we have 10 2 + 4 2 = x 2 , 100 + 16 = x 2 , x 2 = 116 , So x is equal to sqaure root of 116, we get x = 10.77 rounding to the nearest tenth is 11 m.

Question 6. Open-Ended Brianne purchased a box with a base diagonal length that is 45 centimeters long. What are two possible pairs of dimensions for the length and width of the box? Round to the nearest tenth. Answer: One pair: Length = 31.82 cm and Width = 31.82 cm, Other pair: Length = 30 cm and Width = 33.54 cm,

Explanation: Given Brianne purchased a box with a base diagonal length that is 45 centimeters long. The two possible pairs of dimensions for the length and width of the box are the base of the box has a shape of a rectangle, and to find the length of the diagonal we can use the pythagoras theorem as follows: d 2 = l 2 + w 2 ,Where d is the diagonal, l is the length and w is the width. If we assume the base of the box to be a square, we can find one pair of dimension: as l = w, d 2 = l 2 + l 2 = 2l 2 , d 2 = 2 X 45 X 45 = 4,050, d is equal to square root of 4,050 is 63.63, d 2 = 2l 2 , l = 63.63/2  = 31.82 cm, So one pair is length = 31.82 cm and width = 31.82 cm, To find other pair, we can just assume a value for the length, so length = 30 cm then we can find the width as 45 2 = 30 2 + w 2 , w 2 = 2,025 – 900 = 1,125, w is eqaul to square root of 1,125 = 33.54 cm, So the other pair is Length = 30 cm and Width = 33.54 cm.

For Problems 7-9, find the value for the unknown measurement in each rectangular prism or right cone. Round to the nearest tenth.

Explanation: Given measurements as 24 in, 16 in and x in, applying Pythagorean theorem we have 24 2 + 16 2 = x 2 , 576 + 256 = x 2 , x 2 = 832 , So x is equal to sqaure root of 832, we get x = 28.84 rounding to the nearest tenth is 29 in.

Explanation: Given measurements as 40 mm, x mm and 90 mm applying Pythagorean theorem we have 40 2 + x 2 = 90 2 , 1,600 + x 2 = 8,100, x 2 = 8,100 – 1,600 = 6,500 , So x is equal to sqaure root of 6,500, we get x = 80.62 rounding to the nearest tenth is 81 mm.

Explanation: Given measurements as 54 cm, 40 cm and x cm, First finding the hypotenuse side length of 54 cm, 12 cm and y cm, applying Pythagorean theorem we have 54 2 + 12 2 = y 2 , 2,916 + 144 = y 2 , y 2 = 3,060, So y is equal to sqaure root of 3,060, we get y = 55.31 rounding to the nearest tenth is 55 cm. Now we have 55 cm, 40 cm and x cm, applying Pythagorean theorem we have 55 2 + 40 2 = x 2 , 3,025 + 1,600 = x 2 , x 2 = 4,625, So x is equal to sqaure root of 4,625, we get x = 68.0073 rounding to the nearest tenth is 68 cm.

Question 10. Use Structure The Maron Luggage Company claims they have a suitcase large enough to fit a 34-inch baseball bat inside. If the suitcase has height and depth 17 inches by 11 inches, what must its whole-number length be, at minimum? Answer: Its whole- number length will be 34 inches,

Explanation: Given the Maron Luggage Company claims they have a suitcase large enough to fit a 34-inch baseball bat inside. If the suitcase has height and depth 17 inches by 11 inches, Its whole-number length will be 34 inch only aslength will be the same, therefore it is 34 inches.

Explanation: Given cone A – x as 10 cm, y as 17 cm, let z be the height applying Pythagorean theorem we have 10 2 + z 2 = 17 2 , 100 + z 2 = 28, z 2 = 289 – 100 = 189, So z is equal to sqaure root of 189, we get z = 13.74 cm rounding to the nearest tenth is 14 cm, given cone B – x as 12 cm, y as 18 cm, let z be the htpotenuse, applying Pythagorean theorem we have 12 2 + z 2 = 18 2 , 144 + z 2 = 324, z 2 = 324 – 144 = 180, So z is equal to sqaure root of 180, we get z = 13.41 cm rounding to the nearest tenth is 13 cm, upon compairing the heights of Cone A and Cone B , cone A > cone B as 14 cm > 13 cm, therefore cone A is taller than cone B.

B. By how much? Round to the nearest hundredth. Answer: Almost 1 cm,

Explanation: Cone A has height 13.74 cm and cone B has 13.41 cm, So much taller is  13.74 cm – 13.41 = 0.33 cm, nearly 1 cm tall.

Explanation: Given 24 in and 38 in and x in applying Pythagorean theorem we have 24 2 + x 2 = 38 2 , 576 + x 2 =1,444, x 2 = 1,444 – 576 = 868, So x is equal to sqaure root of 868, we get x = 29.46 in rounding to the nearest tenth is 29 in. Therefore the value of x in the diagram is 29 in.

Explanation: Asking to find the length of the longest rod that can fit inside this cube is Length of a cube(l) = 2 m, Breadth of a cube(b) = 2 m, Height of a cube(h) = 2m, The length of the longest rod is the diagonal of the cube. So, We have to find the diagonal of the cube as Diagonal of a cube = √ 3 X side Diagonal of a cube = 1.732 X 2, Diagonal of a cube = 3.46 ≈ 4 m. So the length of the longest rod that can fit inside this cube is 4 m.

Lesson 11.3 More Practice/Homework

Explanation: Given geography the distances between Centerville, Springfield, and Capital City form a right triangle. The distance between Centerville and Springfield is 913 kilometers and the distance between Springfield and Capital City is 976 kilometers. Let x be the direct distance between Centerville and Capital City applying Pythagorean theorem we have 913 2 + 976 2 = x 2 , 833,569 + 952,576 = x 2 , x 2 = 17,86,145, So x is equal to sqaure root of 17,86,145, we get x = 1,336.46 rounding to the nearest tenth is 1,336 kilometers.

B. Anwell travels from Centerville to Springfield, then on to Capital City. Yue travels directly from Centerville to Capital City. How much farther does Anwell travel than Yue? Round to the nearest kilometer. Answer: 553 kilometers,

Explanation: Given Anwell travels from Centerville to Springfield, then on to Capital City. Yue travels directly from Centerville to Capital City. Centerville to Springfield then on to Capital City 913 kilometers + 976 kilometers = 1,889 kilometers, Yue travels directly from Centerville to Capital City is 1,336 kilometers, Anwell travel than Yue is 1,889 kilometers – 1,336 kilometers = 553 kilometers.

Explanation: Given the lengths of ice cup as height 4 in hypotenuse 4.5 in, let r be the radius of  ice cup, applying Pythagorean theorem we have 4 2 + r 2 = 4.5 2 , 16 + r 2 = 20.25, r 2 = 20.25 – 16, So x is equal to sqaure root of 4.25, we get r = 2.06 rounding to the nearest tenth is 2 in.

Explanation: Given a child has an empty box that measures 4 inches by 6 inches by 3 inches. The length of the longest pencil that will fit into the box is the diagonal of cuboid, which is   = √ l 2 + b 2 + h 2 , ∴ Diagonal of box = √ 4 2 + 6 2 + 3 2 , ​ ∴ Diagonal of box =√ 16 + 36 + 9, ∴ Diagonal of box = √61 = 7.81 on rounding it is 8 inches, therefore the lon gest pencil that will fit into the box is 8 inches.

Explanation: Given Mr. Johnston supports a young tree by using a stake and a string forming a right angle with the ground. Let l be the length of the string applying Pythagorean theorem we have 60 2 + 33 2 = l 2 , 3,600 + 1,089 = l 2 , l 2 = 4,689, So l is equal to sqaure root of 4,689, we get l = 68.47 rounding to the nearest tenth is 68 in.

Question 5. The state of Wyoming is almost rectangular, with an approximate width of 365 miles and an approximate height of 276 miles. If you fly across the state from opposite corners, what is the best approximation of the distance you travel? (A) 448 miles (B) 450 miles (C) 458 miles (D) 462 miles Answer: (C) 458 miles,

Explanation: Given the state of Wyoming is almost rectangular with an approximate width of 365 miles and an approximate height of 276 miles. If we fly across the state from opposite corners, the best approximation of the distance I travel is let it be d applying Pythagorean theorem we have 365 2 + 276 2 = d 2 , 132,225 + 76,176 = d 2 , d 2 = 209,401, So d is equal to sqaure root of 209,401 we get d = 457.60 rounding to the nearest tenth is 458 miles which matches with bit (C).

Explanation: Given a box measuring 100 centimeters by 130 centimeters by 400 centimeters The length of the longest rod that will fit into the box is the diagonal of cuboid, which is   = √ l 2 + b 2 + h 2 , ∴ Diagonal of box = √ 100 2 + 130 2 + 400 2 , ​ ∴ Diagonal of box =√ 10,000 + 16,900 + 160,000, ∴ Diagonal of box = √186,900= 432.31 on rounding it is 432 centimeters, therefore the longest rod that will fit into the box is 432 centimeters.

Explanation: The measurements that could possibly fit the given cone diagram as it is a right traingle we applying Pythagorean theorem and checking with bits as Bit (A) x = 3, y = 4, z = 5, 3 2 + 4 2 = 5 2 , 9 + 16 = 25   as 25 = 25 matches, checking with bit (B) x = 9, y = 6, z = 10, 9 2 + 6 2 = 81 + 36 = 117 ≠ 10 2 ≠ 100, So not form a right triangle, Checking with bit (C) x = 3, y = 8, z = 5, 3 2 + 8 2 = 9 + 64 = 73 ≠ 5 2 ≠ 25, So not form a right triangle, Checking with bit (D) x = 6, y = 10, z = 8, 6 2 + 10 2 = 36 + 100 = 136 ≠ 8 2 ≠ 64, So not form a right triangle, Therfore only bit (A) x = 3, y = 4, z = 5 measurements that could possibly fit.

Spiral Review

Question 8. A fun park charges $10 for admission and then $2 for every ride. Write a function that determines the amount of money spent, y, based on the number of rides ridden, x. Answer: y = 2x + 10,

Explanation: Given a park charges $10 for admission and then $2 for every ride. The function that determines the amount of money spent y based on the number of rides ridden x is y= 2x + 10.

Explanation: Reflected the triangle across the y-axis and drawn the image as shown above, Given A(-5,2), B(-7,6) and C(-3,8) so we get A(-5,-2), B(-7,-6) and C(-3,-8).

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