problem solving find unknown lengths lesson 7 10

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Go Math Interactive Mimio Lesson 7.10 Problem Solving Find Unknown Lengths

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Polygons - Lesson 11.1

Triangles - Lesson 11.2

Quadrilaterals - Lesson 11.3

Three Dimensional Figures - Lesson 11.5

Unit Cubes and Solid Figures - Lesson 11.6

Understanding Volume - Lesson 11.7

Estimate Volume - Lesson 11.8

Volume of a Rectangular Prism - Lesson 11.9

Apply Volume Formulas - Lesson 11.10

Finding Volume of Composite Formulas - Lesson 11.12

Find a Part of a Group - Lesson 7.1

Multiply Fractions and Whole Numbers - Lesson 7.2

Fraction and Whole Number Multiplication - Lesson 7.3

Multiply Fractions - Lesson 7.4

Compare Fraction Factor and Product - Lesson 7.5

Fraction Multiplication - Lesson 7.6

Area and Mixed Numbers - Lesson 7.7

Compare Mixed Number Factors and Products - Lesson 7.8

Multiply Mixed Numbers - Lesson 7.9

Problem Solving - Find Unknown Lengths - Lesson 7.10  

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Line Plots - Lesson 9.1

Ordered Pairs - Lesson 9.2

Graph Data - Lesson 9.3

Line Graphs - Lesson 9.4

Numerical Patterns - Lesson 9.5

Problem Solving - Find a Rule - Lesson 9.6

Graph and Analyze Relationships - Lesson 9.7

Customary Length - Lesson 10.1

Customary Capacity - Lesson 10.2

Weight - Lesson 10.3

Multistep Measurement Problems - Lesson 10.4

Metric Measures - Lesson 10.5

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Division Patterns with Decimals - Lesson 5.1

Divide Decimals by Whole Numbers - Lesson 5.2

Estimate Quotients - lesson 5.3

Division of Decimals by Whole Numbers - Lesson 5.4

Decimal Division - Lesson 5.5

Divide Decimals - Lesson 5.6

Write Zeros in the Dividend - Lesson 5.7

Problem Solving - Decimal Operations - Lesson 5.8

Divide Fractions and Whole Numbers - Lesson 8.1

Problem Solving - Use Multiplication - Lesson 8.2

Connect Fractions to Division - Lesson 8.3

Fraction and Whole Number Division - Lesson 8.4

Interpret Division with Fractions - Lesson 8.5

Addition with Unlike Denominators - Lesson 6.1

Subtraction with Unlike Denominators - Lesson 6.2

Estimate Fraction Sums and Differences - Lesson 6.3

Common Denominators and Equivalent Fractions - Lesson 6.4

Add or Subtract Fractions - Lesson 6.5

Add or Subtract Mixed Numbers - Lesson 6.6

Subtraction with Renaming - Lesson 6.7

Patterns with Fractions - Lesson 6.8

Problem Solving with Addition and Subtraction - Lesson 6.9

Use Properties of Addition - Lesson 6.10

Multiplication Patterns with Decimals - Lesson 4.1

Multiply Decimals and Whole Numbers - Lesson 4.2

Multiply Decimals and Whole Numbers - Lesson 4.3

Multiply Using Expanded Form - Lesson 4.4

Problem Solving - Multiply Money - Lesson 4.5

Decimal Multiplication - Lesson 4.6

Multiply Decimals - Lesson 4.7

Thousandths - Lesson 3.1

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Estimate Decimal Sums and Differences - Lesson 3.7

Add Decimals - Lesson 3.8

Subtract Decimals - Lesson 3.9

Patterns with Decimals - Lesson 3.10

Problem Solving Add and Subtract Money - Lesson 3.11

Choose a Method - Lesson 3.12

Performance Task on Chapter 3

Place the First Digit - Lesson 2.1

Divide by 1-Digit Divisors - Lesson 2.2

Division with 2-Digit Divisors - Lesson 2.3

Partial Quotients - Lesson 2.4

Estimate with 2-Digit Divisors - Lesson 2.5

Divide by 2-Digit Divisors - Lesson 2.6

Interpret the Remainder - Lesson 2.7

Adjust Quotients - Lesson 2.8

Problem Solving - Division - Lesson 2.9

Performance Task on Chapter 2

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8.2.3: Finding Unknown Side Lengths

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Let's find missing side lengths of right triangles.

Exercise \(\PageIndex{1}\): Which One Doesn't Belong: Equations

Which one doesn't belong?

\(3^{2}+b^{2}=5^{2}\)

\(b^{2}=5^{2}-3^{2}\)

\(3^{2}+5^{2}=b^{2}\)

\(3^{2}+4^{2}=5^{2}\)

Exercise \(\PageIndex{2}\): Which One is the Hypotenuse?

Label all the hypotenuses with \(c\).

Exercise \(\PageIndex{3}\): Find the Missing Side Lengths

1. Find \(c\).

2. Find \(b\).

3. A right triangle has sides of length 2.4 cm and 6.5 cm. What is the length of the hypotenuse?

4. A right triangle has a side of length \(\frac{1}{4}\) and a hypotenuse of length \(\frac{1}{3}\). What is the length of the other side?

5. Find the value of \(x\) in the figure.

Are you ready for more?

The spiral in the figure is made by starting with a right triangle with both legs measuring one unit each. Then a second right triangle is built with one leg measuring one unit, and the other leg being the hypotenuse of the first triangle. A third right triangle is built on the second triangle’s hypotenuse, again with the other leg measuring one unit, and so on.

Find the length, \(x\), of the hypotenuse of the last triangle constructed in the figure.

There are many examples where the lengths of two legs of a right triangle are known and can be used to find the length of the hypotenuse with the Pythagorean Theorem. The Pythagorean Theorem can also be used if the length of the hypotenuse and one leg is known, and we want to find the length of the other leg. Here is a right triangle, where one leg has a length of 5 units, the hypotenuse has a length of 10 units, and the length of the other leg is represented by \(g\).

Start with \(a^{2}+b^{2}=c^{2}\), make substitutions, and solve for the unknown value. Remember that \(c\) represents the hypotenuse: the side opposite the right angle. For this triangle, the hypotenuse is 10.

\[\begin{aligned} a^{2}+b^{2}&=c^{2}\\5^{2}+g^{2}&=10^{2}\\g^{2}&=10^{2}-5^{2}\\g^{2}&=100-25\\g^{2}&=75\\g&=\sqrt{25}\end{aligned}\nonumber\]

Use estimation strategies to know that the length of the other leg is between 8 and 9 units, since 75 is between 64 and 81. A calculator with a square root function gives \(\sqrt{75}\approx 8.66\).

Glossary Entries

Definition: Pythagorean Theorem

The Pythagorean Theorem describes the relationship between the side lengths of right triangles.

The diagram shows a right triangle with squares built on each side. If we add the areas of the two small squares, we get the area of the larger square.

The square of the hypotenuse is equal to the sum of the squares of the legs. This is written as \(a^{2}+b^{2}=c^{2}\).

Definition: Hypotenuse

The hypotenuse is the side of a right triangle that is opposite the right angle. It is the longest side of a right triangle.

Here are some right triangles. Each hypotenuse is labeled.

Definition: LEgs

The legs of a right triangle are the sides that make the right angle.

Here are some right triangles. Each leg is labeled.

Exercise \(\PageIndex{4}\)

Find the exact value of each variable that represents a side length in a right triangle.

Exercise \(\PageIndex{5}\)

A right triangle has side lengths of \(a\), \(b\), and \(c\) units. The longest side has a length of \(c\) units. Complete each equation to show three relations among \(a\), \(b\), and \(c\).

(From Unit 8.2.2)

Exercise \(\PageIndex{6}\)

What is the exact length of each line segment? Explain or show your reasoning. (Each grid square represents 1 square unit.)

Exercise \(\PageIndex{7}\)

In 2015, there were roughly \(1\times 10^{6}\) high school football players and \(2\times 10^{3}\) professional football players in the United States. About how many times more high school football players are there? Explain how you know.

(From Unit 7.3.7)

Exercise \(\PageIndex{8}\)

• \(\left(\frac{1}{2}\right)^{3}\)
• \(\left(\frac{1}{2}\right)^{-3}\)

(From Unit 7.2.5)

Exercise \(\PageIndex{9}\)

Here is a scatter plot of weight vs. age for different Dobermans. The model, represented by \(y=2.45x+1.22\), is graphed with the scatter plot. Here, \(x\) represents age in weeks, and represents weight in pounds.

• What does the slope mean in this situation?
• Based on this model, how heavy would you expect a newborn Doberman to be?

(From Unit 6.2.4)

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