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1.6: Points that Partition Line Segments

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Section formula finds coordinates of a point that splits a line segment in a given ratio.

Recall that a median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the side opposite the vertex. All triangles have three medians and these three medians intersect in one point called the centroid , shown below. The centroid partitions each median in a 2:1 ratio.

Find the coordinates of the centroid, given the coordinates of the vertices of the triangle as shown.

Suppose you have a line segment \(\overline{AB}\). A point \(P\) divides this line segment into two parts such that \(AP=mk\) and \(PB=nk\). You can say that point \(P\) partitions segment \(AB\) in a \(m:n\) ratio. (Note that \(\dfrac{mk}{nk}=mn\), a ratio of \(m:n\).)

A natural question to ask is, what are the coordinates of point \(P\)? It turns out that with the help of similar triangles and algebra, you can come up with a formula that will give you the coordinates of point \(P\) based on the coordinates of \(A\), the coordinates of \(B\), and the ratio \(m:n\). This formula is sometimes referred to as the section formula .

Section Formula: Given \(\overline{AB}\) with \(A=(x_1, y_1)\) and \(B=(x_2, y_2)\), if point \(P\) partitions \(\overline{AB}\) in a \(m:n\) ratio, then the coordinates of point P are:

\(P=\left (\dfrac{mx_{2}+nx_{1}}{m+n}, \dfrac{my_{2}+ny_{1}}{m+n}\right )\)

Proving Triangle Similarity

For segment \(\overline{AB}\) below, draw two right triangles, one with hypotenuse \(\overline{AP}\) and one with hypotenuse \(\overline{PB}\). Show that these triangles are similar.

Start by drawing the right triangles. Below, the base and height of each triangle has been labeled in green.

Clearly these triangles have one pair of congruent angles (the right angles). What other information do you have about the triangles? You know that for each triangle, the ratio \(\dfrac{height}{base}\) is the slope of \(\overline{AB}\). Because these two triangles are attached to the same line segment with the same slope, it means that \(\dfrac{h_1}{b_1} = {h_2}{b_2}\). This is equivalent to \(\dfrac{b_2}{b_1}=\dfrac{h_2}{h_1}\). Two pairs of sides are in the same ratio.

Not only is there one pair of congruent angles, but there are also two pairs of corresponding sides with the same ratio. The triangles are similar by \(SAS\)∼.

Finding Length and Height

Find the lengths of the bases and heights of each triangle. Use the fact that the triangles are similar to set up and solve proportions for \(x\) and then for \(y\) in order to find the coordinates of point \(P\).

The bases and heights can be found in terms of \(x_1\), \(y_1\), \(x\), \(y\), \(x_2\), \(y_2\).

Because the triangles are similar, the ratios between pairs of corresponding sides are equal. In particular, you know:

• \(\dfrac{mk}{nk}=\dfrac{x−x_1}{x_2−x}\)
• \(\dfrac{mk}{nk}=\dfrac{y−y_1}{y_2−y}\)

You can use algebra to solve the first equation for \(x\) and the second equation for \(y\).

• \(\dfrac{mk}{nk}=\dfrac{x−x_1}{x_2−x}\) \(\rightarrow \dfrac{m}{n}=\dfrac{x−x_1}{x_2−x} \)

\(\rightarrow mx_2−mx=nx−nx_1\)

\(\rightarrow mx_2+mx_1=mx+nx \)

\(\rightarrow mx_2−nx_1=x(m+n)\)

\(\rightarrow \dfrac{mx_2−nx_1}{m+n}=\dfrac{x(m+n)}{m+n}\)

\(\rightarrow x=\dfrac{mx_2+nx_1}{m+n}\)

• \(\dfrac{mk}{nk}=\dfrac{y−y_1}{y_2−y}\) \(\rightarrow\dfrac{m}{n}=\dfrac{y−y_1}{y_2−y} \)

\(\rightarrow my_2−my=ny−ny_1\)

\(\rightarrow my_2+my_1=my+ny\)

\(\rightarrow my_2−ny_1=y(m+n)\)

\(\rightarrow \dfrac{my_2−ny_1}{m+n}=\dfrac{y(m+n)}{m+n}\)

\(\rightarrow x=\dfrac{my_2+ny_1}{m+n}\)

Point \(P\) is at:

\(P=\left(\dfrac{mx_2+nx_1}{m+n} ,\dfrac{my_2+ny_1}{m+n}\right)\)

Finding the Coordinates of a Point

Consider \(\overline{AB}\) with \(A=(10, 2)\) and \(B=(4, 1)\). \(P\) partitions \(\overline{AB}\) in a ratio of \(2:3\). Find the coordinates of point \(P\).

You can use the section formula with \((x_1,y_1)=(10, 2)\), \((x_2, y_2)=(4, 1)\), \(m=2\), \(n=3\).

\(P=\left(\dfrac{mx_2+nx_1}{m+n} ,\dfrac{my_2+ny_1}{m+n}\right)=\left(\dfrac{2 \cdot 4+3 \cdot10}{2+3}, \dfrac{2 \cdot 1+3\cdot 2}{2+3}\right)=\left(7.6,1.6\right)\)

You can plot points \(A\), \(B\), and \(P\) to see if this answer is realistic.

This does look like \(P\) partitions the segment from \(A\) to \(B\) in a ratio of \(2:3\). Note that the answer would be different if you were looking for the point that partitioned the segment from \(B\) to \(A\). The order of the letters and “direction” of the segment matters.

Example \(\PageIndex{1}\)

Earlier, you were asked to find the coordinates of the centroid, given the coordinates of the vertices of the triangle as shown.

One way to find the coordinates of the centroid is to use the section formula. You can focus on any of the three medians. Here, look at the median from point A. First, you will need to find the coordinates of the midpoint of \(\overline{BC}\) (the midpoint formula , a special case of the section formula, is derived in Guided Practice #1 and #2):

\(\left(\dfrac{x_2+x_1}{2}, \dfrac{y_2+y_1}{2}\right)=\left (\dfrac{5+6}{2}, \dfrac{5+1}{2}\right)=\left(5.5,3\right )\)

\(\left(\dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n}\right)=\left(\dfrac{2 \cdot 5.5+1 \cdot 2}{2+1}, \dfrac{2 \cdot 3+1 \cdot 6}{2+1}\right)=\left(\dfrac{13}{3}, 4\right)\)

Looking at the picture, these coordinates for the centroid are realistic.

Example \(\PageIndex{2}\)

The midpoint of a line segment is the point exactly in the middle of the line segment. In what ratio does a midpoint partition a segment?

1:1, because the segments connecting the midpoint to each endpoint will be the same length.

Example \(\PageIndex{3}\)

The midpoint formula is a special case of the section formula where m=n=1. Derive a formula that calculates the midpoint of the segment connecting \((x_1,y_1)\) with \((x_2, y_2)\).

For a midpoint, \(m=n=1\). The section formula becomes:

\(\left(\dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n}\right)=\left(\dfrac{x_2+x_1}{2}, \dfrac{y_2+y_1}{2}\right)

This is the midpoint formula.

Example \(\PageIndex{4}\)

Consider \(\overline{BA}\) with \(B=(4, 1)\) and \(A=(10, 2)\). \(P\) partitions the segment in a ratio of \(2:3\). Find the coordinates of point \(P\). How and why is this answer different from the answer to Example \(C\)?

\((x_1, y_1)=(4,1)\) and \((x_2, y_2)=(10,2)\). \(m=2\) and \(n=3\).

\(P=\left(\dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n}\right)=\left(\dfrac{2 \cdot 10+3 \cdot 4}{2+3}, \dfrac{2 \cdot 2+3 \cdot 1}{2+3}\right)=\left(6.4,1.4\right)\)

This answer is different from the answer to Example C because in this case point P is partitioning the segment in a \(2:3\) ratio starting from point \(B\). In Example \(C\), you were starting from point \(A\).

Find the midpoint of each of the following segments defined by the given endpoints.

1. \((2, 6)\) and \((−4, 8)\)

2. \((1, 9)\) and \((−2, 5)\)

3. \((11, 24)\) and \((8, 12)\)

4. \((1, 3)\) is the midpoint of \(\overline{AB}\) with \(A=(−2, 1)\). Find the coordinates of \(B\).

5. \((2, 4)\) is the midpoint of \(\overline{CD}\) with \(C=(−5, 9)\). Find the coordinates of \(D\).

6. \((4, 23)\) is the midpoint of \(\overline{EF}\) with \(E=(7, 11)\). Find the coordinates of \(F\).

Consider \(A=(−9, 4)\) and \(B=(11, 17)\).

7. Point \(P_1\) partitions the segment from \(A\) to \(B\) in a \(3:5\) ratio. Find the coordinates of point \(P_1\).

8. Point \(P_2\) partitions the segment from \(B\) to \(A\) in a \(3:5\) ratio. Find the coordinates of point \(P_2\).

9. Why are the answers to 7 and 8 different?

10. Find the length of \(AP_1\) and \(P_2B\). Why should these lengths be the same?

Consider \(C=(−6, −1)\) and \(D=(4, 8)\).

11. Point \(P_3\) partitions the segment from \(A\) to \(B\) in a \(1:2\) ratio. Find the coordinates of point \(P_3\).

12. Point \(P_4\) partitions the segment from \(B\) to \(A\) in a \(4:5\) ratio. Find the coordinates of point \(P_4\).

13. Point \(P=(1, 2)\) partitions the segment from \(E=(9, 6)\) to \(F\) in a \(2:5\) ratio. Find the coordinates of point F.

14. Point \(P=(−6, −4)\) partitions the segment from \(G=(−4, 6)\) to \(H\) in a \(5:3\) ratio. Find the coordinates of point \(H\).

15. Point \(P=(6,8)\) partitions the segment from \(I=(−2,1)\) to \(J\) in a \(6:7\) ratio. Find the coordinates of point \(J\).

16. A triangle is defined by the points \((5, 6)\), \((9, 17)\), and \((−2, 1)\). Find the coordinates of the centroid of the triangle.

Review (Answers)

To see the Review answers, open this PDF file and look for section 10.6.

Additional Resource

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Practice: Points that Partition Line Segments

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The Art of Partitioning a Line Segment!

In geometry, the simple act of drawing a line segment can lead to deeper, fascinating explorations. One such exploration is the partitioning of a line segment. Imagine dividing a chocolate bar into pieces, ensuring each friend gets an equal share, or maybe one gets twice as much as another. Similarly, in geometry, we can split a line segment into multiple parts based on a specific ratio. Let's delve into the process of partitioning a line segment and understand its mathematical underpinnings.

Step-by-step Guide: Partitioning a Line Segment

Understanding the Concept : Partitioning a line segment involves dividing it into multiple parts, where each part is a fraction or multiple of the whole segment. This can be done based on a given ratio.

Mathematical Representation : Given a line segment \(AB\), we can partition it at a point \(P\) such that the ratio of \(AP\) to \(PB\) is \(m:n\), where \(m\) and \(n\) are positive integers.

• Formula for Partitioning : If \(A(x_1, y_1)\) and \(B(x_2, y_2)\) are the endpoints of the segment, and we want to partition the segment in the ratio \(m:n\), the coordinates \((x, y)\) of point \(P\) are given by: \( x = \frac{mx_2 + nx_1}{m+n} \) \( y = \frac{my_2 + ny_1}{m+n} \)

Example 1 Given the line segment with endpoints \(A(1,2)\) and \(B(7,8)\), find the point that partitions the segment in the ratio \(2:3\).

Solution : Using the formula: \( x = \frac{2 \times 7 + 3 \times 1}{2+3} = \frac{17}{5} = 3.4 \) \( y = \frac{2 \times 8 + 3 \times 2}{2+3} = \frac{22}{5} = 4.4 \) Thus, the required point is \(P(3.4, 4.4)\).

Example 2 : Partition the line segment with endpoints \(C(3,4)\) and \(D(9,12)\) in the ratio \(1:4\).

Solution : Applying the formula: \( x = \frac{1 \times 9 + 4 \times 3}{1+4} = \frac{21}{5} = 4.2 \) \( y = \frac{1 \times 12 + 4 \times 4}{1+4} = \frac{28}{5} = 5.6 \) The partition point is \(P(4.2, 5.6)\).

Practice Questions :

• For the line segment with endpoints \(E(2,3)\) and \(F(10,7)\), determine the point that partitions the segment in the ratio \(3:2\).
• Partition the line segment with endpoints \(G(-1,2)\) and \(H(5,10)\) in the ratio \(4:1\).
• \(P(6.8,5.4)\)
• \(P(3.8,8.4)\)

by: Effortless Math Team about 6 months ago (category: Articles )

Effortless Math Team

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Partitioning a Segment in a Given Ratio

Suppose you have a line segment P Q ¯ on the coordinate plane, and you need to find the point on the segment 1 3  of the way from P to Q .

Let’s first take the easy case where P is at the origin and line segment is a horizontal one.

The length of the line is 6 units and the point on the segment 1 3  of the way from P to Q would be 2 units away from P , 4 units away from Q and would be at ( 2 , 0 ) .

Consider the case where the segment is not a horizontal or vertical line.

The components of the directed segment P Q ¯ are 〈 6 , 3 〉  and we need to find the point, say X on the segment 1 3  of the way from P to Q .

Then, the components of the segment P X ¯ are 〈 ( 1 3 ) ( 6 ) , ( 1 3 ) ( 3 ) 〉 = 〈 2 , 1 〉 .

Since the initial point of the segment is at origin, the coordinates of the point X are given by ( 0 + 2 , 0 + 1 ) = ( 2 , 1 ) .

Now let’s do a trickier problem, where neither P nor Q is at the origin.

Use the end points of the segment P Q ¯  to write the components of the directed segment.

〈 ( x 2 − x 1 ) , ( y 2 − y 1 ) 〉 = 〈 ( 7 − 1 ) , ( 2 − 6 ) 〉                                                                                         = 〈 6 , − 4 〉

Now in a similar way, the components of the segment P X ¯  where X is a point on the segment 1 3  of the way from P to Q are 〈 ( 1 3 ) ( 6 ) , ( 1 3 ) ( − 4 ) 〉 = 〈 2 , − 1.25 〉 .

To find the coordinates of the point X add the components of the segment P X ¯  to the coordinates of the initial point P .

So, the coordinates of the point X are ( 1 + 2 , 6 − 1.25 ) = ( 3 , 4.75 ) .

Note that the resulting segments, P X ¯ and X Q ¯ , have lengths in a ratio of 1 : 2 .

In general: what if you need to find a point on a line segment that divides it into two segments with lengths in a ratio a : b ?

Consider the directed line segment X Y ¯  with coordinates of the endpoints as X ( x 1 , y 1 )  and Y ( x 2 , y 2 ) .

Suppose the point Z divided the segment in the ratio a : b , then the point is a a + b of the way from X to Y .

So, generalizing the method we have, the components of the segment X Z ¯ are 〈 ( a a + b ( x 2 − x 1 ) ) , ( a a + b ( y 2 − y 1 ) ) 〉 .

Then, the X -coordinate of the point Z is

x 1 + a a + b ( x 2 − x 1 ) = x 1 ( a + b ) + a ( x 2 − x 1 ) a + b                                                                                 = b x 1 + a x 2 a + b .

Similarly, the Y -coordinate is

y 1 + a a + b ( y 2 − y 1 ) = y 1 ( a + b ) + a ( y 2 − y 1 ) a + b                                                                                 = b y 1 + a y 2 a + b .

Therefore, the coordinates of the point Z are ( b x 1 + a x 2 a + b , b y 1 + a y 2 a + b ) .

Find the coordinates of the point that divides the directed line segment M N ¯ with the coordinates of endpoints at M ( − 4 , 0 )  and M ( 0 , 4 ) in the ratio 3 : 1 ?

Let L be the point that divides M N ¯  in the ratio 3 : 1 .

Here, ( x 1 , y 1 ) = ( − 4 , 0 ) , ( x 2 , y 2 ) = ( 0 , 4 ) and a : b = 3 : 1 .

Substitute in the formula. The coordinates of L are

( 1 ( − 4 ) + 3 ( 0 ) 3 + 1 , 1 ( 0 ) + 3 ( 4 ) 3 + 1 ) .

( − 4 + 0 4 , 0 + 12 4 ) = ( − 1 , 3 )

Therefore, the point L ( − 1 , 3 )  divides M N ¯  in the ratio 3 : 1 .

What are the coordinates of the point that divides the directed line segment A B ¯  in the ratio 2 : 3 ?

Let C be the point that divides A B ¯  in the ratio 2 : 3 .

Here, ( x 1 , y 1 ) = ( − 4 , 4 ) , ( x 2 , y 2 ) = ( 6 , − 5 ) and a : b = 2 : 3 .

Substitute in the formula. The coordinates of C are

( 3 ( − 4 ) + 2 ( 6 ) 5 , 3 ( 4 ) + 2 ( − 5 ) 5 ) .

( − 12 + 12 5 , 12 − 10 5 ) = ( 0 , 2 5 )                                                                                         = ( 0 , 0.4 )

Therefore, the point C ( 0 , 0.4 )  divides A B ¯  in the ratio 2 : 3 .

You can note that the Midpoint Formula is a special case of this formula when a = b = 1 .

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