Topics
12. Connecting Algebra and Geometry Through Coordinates
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United States of America
Grades 9-12

12.01 Dividing segments

Lesson
Practice

Introduction

We began working with ratios in 6th grade, and we have used them more recently with similar triangles in lesson  7.02 Similarity transformations  . In this lesson, we will partition line segments into a given ratio using formulas and similar triangles.

Midpoints

We can divide a line segment into two congruent pieces by finding the midpoint.

Exploration

Three line segments are graphed below.

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  1. Find the midpoint of each of the segments.
  2. Describe your strategy for finding the midpoint of \left(2,3\right) and \left(6,11\right).
  3. Think of and describe a second strategy for finding the midpoint of \left(2,3\right) and \left(6,11\right).
  4. How are the two strategies related?

We can find the midpoint of a line segment using the following formula, which shows that the coordinates of the midpoint are the average of the coordinates of the endpoints.

\displaystyle \left(x_M,y_M\right)=\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} \right)
\bm{\left(x_M,y_M\right)}
coordinates of the midpoint
\bm{\left(x_1,y_1\right)}
coordinates of the first endpoint
\bm{\left(x_2,y_2\right)}
coordinates of the second endpoint

Another strategy for finding the midpoint is to use similar triangles. We can count squares or use absolute values to find the lengths of the legs, the rise and run, of the right triangle with hypotenuse \overline{AB}. Then, we can create a similar triangle that is \frac{1}{2} the size of the original triangle.

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Original triangle with hypontenuse \overline{AB}
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New triangle with hypontenuse \frac{1}{2}\left(\overline{AB}\right)

Examples

Example 1

Consider the line segment with endpoints A \left(-5,12\right) and B \left(10,-4\right).

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Find the midpoint, M, of \overline{AB}.

Worked Solution
Create a strategy

For this problem, we will use similar triangles to solve. Using this method, we create a triangle with \overline{AB} as the hypotenuse, find the lengths of the legs of the triangle, then create a similar triangle that is half the size of the original.

Apply the idea
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Using the absolute value to find the lengths of the legs, we get:

\text{Leg } 1=\left|-5-10\right|=15

\text{Leg }2=\left|12-\left(-4\right)\right|=16

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The legs of the new triangle need to be half the size of the legs of the original triangle:

\text{New run}=\frac{1}{2}\cdot 15=7.5

\text{New rise}=\frac{1}{2}\cdot 16=8

From point A\left(-5,12\right) moving right 7.5 units and down 8 units, we get the midpoint:

M=\left(-5+7.5, 12-8\right)=\left(2.5,4\right)

Reflect and check

We could have used the midpoint formula to find the answer instead, but we can also use it to check our answer.

\displaystyle M\displaystyle =\displaystyle \left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} \right)Formula for midpoint
\displaystyle =\displaystyle \left(\dfrac{-5+10}{2}, \dfrac{12+-4}{2} \right)Substitute coordinates of the endpoints
\displaystyle =\displaystyle \left(\dfrac{5}{2}, \dfrac{8}{2} \right)Evaluate the numerators
\displaystyle =\displaystyle \left(2.5, 4 \right)Evaluate the division

Example 2

The midpoint of A \left(1,4\right) and B \left(a,b\right) is M\left(9,7\right).

a

Find the value of a.

Worked Solution
Create a strategy

We can solve for the missing endpoint by substituting the midpoint and known endpoint into the formula for midpoint. To find a, we only need to look at the x-coordinate.

Apply the idea
\displaystyle M\left(x_M, y_M\right)\displaystyle =\displaystyle \left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} \right)Formula for midpoint
\displaystyle x_M\displaystyle =\displaystyle \dfrac{x_1+x_2}{2}Only the x-coordinate
\displaystyle 9\displaystyle =\displaystyle \dfrac{1+a}{2}Substitute the x-coordinates
\displaystyle 18\displaystyle =\displaystyle 1+aMultiply both sides by 2
\displaystyle 17\displaystyle =\displaystyle aSubtract 1 from both sides
\displaystyle a\displaystyle =\displaystyle 17Symmetric property of equality
b

Find the value of b.

Worked Solution
Create a strategy

To find b, we only need to look at the y-coordinate.

Apply the idea
\displaystyle M\left(x_M, y_M\right)\displaystyle =\displaystyle \left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} \right)Formula for midpoint
\displaystyle y_M\displaystyle =\displaystyle \dfrac{y_1+y_2}{2}Only the y-coordinate
\displaystyle 7\displaystyle =\displaystyle \dfrac{4+b}{2}Substitute the y-coordinates
\displaystyle 14\displaystyle =\displaystyle 4+bMultiply both sides by 2
\displaystyle 10\displaystyle =\displaystyle bSubtract 4 from both sides
\displaystyle b\displaystyle =\displaystyle 10Symmetric property of equality
Reflect and check

This means that the coordinates of B are \left(17,10\right). We can check that this answer is correct by graphing \overline{AB} and using similar triangles.

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The values of the rise and run of the triangle whose hypotenuse is \overline{AB} are 6 and 16 respectively.

\dfrac{1}{2}\cdot \text{run}=\dfrac{1}{2}\cdot 16=8

\dfrac{1}{2}\cdot \text{rise}=\dfrac{1}{2}\cdot 6=3

From point A\left(1,4\right), moving right 8 and up 3, we get:

\left(1+8,4+3\right)=\left(9,7\right)

This shows the given midpoint is halfway between A and B which means our coordinates for B were correct.

Example 3

There is a stretch of nearly perfectly straight road between Dalhart, Texas and Liberal, Kansas along US-54 W.

Dalhart, Texas is located at coordinates of \left(36.06 \degree \text{N}, -102.52 \degree \text{W}\right) and Liberal, Kansas is located at \left(37.05 \degree \text{N}, -100.92 \degree \text{W}\right). Seven cities on this route are given with their coordinates. State which city is closest to the midpoint between Dalhart and Liberal.

  • Conlen, \left(36.24 \degree \text{N}, -102.23 \degree \text{W}\right)
  • Stratford, \left(36.34 \degree \text{N}, -102.07 \degree \text{W}\right)
  • Texhoma, \left(36.51 \degree \text{N}, -101.78 \degree \text{W}\right)
  • Goodwell, \left(36.60 \degree \text{N}, -101.64 \degree \text{W}\right)
  • Guymon, \left(35.68 \degree \text{N}, -101.48 \degree \text{W}\right)
  • Optima, \left(36.76 \degree \text{N}, -101.35 \degree \text{W}\right)
  • Tyrone, \left(36.96 \degree \text{N}, -101.07 \degree \text{W}\right)
Worked Solution
Create a strategy

We can find the midpoint algebraically and then check the map to see which city is closest and ensure we get a reasonable answer.

Apply the idea

Finding the midpoint between: Dalhart, Texas\left(36.06 \degree \text{N}, -102.52 \degree \text{W}\right) and Liberal, Kansas\left(37.05 \degree \text{N}, -100.92 \degree \text{W}\right)

\displaystyle M\displaystyle =\displaystyle \left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)Formula for midpoint
\displaystyle M\displaystyle =\displaystyle \left(\dfrac{36.06+37.05}{2}, \dfrac{-102.52+(-100.92)}{2}\right)Substitute the coordinates
\displaystyle M\displaystyle =\displaystyle \left(36.555, -101.72\right)Evaluate the addition and division

The two cities that are close to this midpoint are:

  • Texhoma, \left(36.51 \degree \text{N}, -101.78 \degree \text{W}\right)
  • Goodwell, \left(36.60 \degree \text{N}, -101.64 \degree \text{W}\right)

The x-coordinates are both 0.045 \degree away, but for Texhoma, the y-coordinate is 0.6 \degree away, while Goodwell is 0.8 \degree away. This means that Texhoma is closer to the halfway mark between Dalhart and Liberal.

Reflect and check

Using the map, we could estimate that it would have been one of Texhoma and Goodwell, but we need to work algebraically to confirm which is closer.

Idea summary

We can find the midpoint of a line segment using the midpoint formula:

\displaystyle M=\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} \right)
\bm{M}
coordinates of the midpoint
\bm{\left(x_1,y_1\right)}
coordinates of the first endpoint
\bm{\left(x_2,y_2\right)}
coordinates of the second endpoint

or using similar triangles. To create a triangle that is half the size of the original, we find:

\left(\frac{1}{2}\right)\cdot\text{rise}\\ \left(\frac{1}{2}\right)\cdot\text{run}

and add those values to the coordinates of the leftmost point.

Partitioning segments

The midpoint of a segment divides the segment in a 1:1 ratio. This means that the line is divided into 2 equal pieces with the first part of the segment made up of one of those pieces, and the second part of the segment made up of the other piece.

If we divide the segment in a 1:2 ratio, that means there are 3 equal pieces with the first part of the segment made up of one of those pieces, and the second part of the segment made up of 2 of those pieces.

The partition point, P, is \frac{1}{3} of the distance from point A to point B.

Exploration

Consider the line segment below.

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  1. Find a point that divides the segment in a 3:5 ratio and describe your process.
  2. Find a point that divides the segment in a 1:3 ratio and describe your process.
  3. How could you apply your process to divide a line segment from \left(0,0\right) to \left(6,3\right) into a 2:1 ratio?

One strategy for partitioning a segment into a m:n ratio is to use similar triangles. For example, consider partitioning segment AB into a 3:2 ratio. This means there are 5 equal pieces with the first part of the segment made up of 3 pieces and the second part of the segment made up of 2 pieces.

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We can count squares or use absolute values to find the lengths of the legs (the rise and run) of the right triangle with hypotenuse \overline{AB}

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We can create a similar triangle that is \frac{3}{5} the size of the original triangle.

\frac{3}{5}\cdot \text{run}=\frac{3}{5}\cdot 5=3

\frac{3}{5}\cdot \text{rise}=\frac{3}{5}\cdot 10=6

From point A\left(2,1\right), moving right 3 and up 6, we get:

\left(2+3,1+6\right)=\left(5,7\right)

We can say that P divides \overline{AB} in the ratio 3:2. Because \overline{AP} is made up of 3 equal pieces and \overline{PB} is made up of 2 equal pieces, point P will be further away from point A.

Examples

Example 4

Consider the line segment with endpoints A \left(-5,-4\right) and B \left(1,8\right).

Find the point, P, that is \frac{1}{3} of the way from A to B.

Worked Solution
Create a strategy

If P is \frac{1}{3} of the way from A to B, then the vertical distance from A to P is \frac{1}{3} of the rise and the horizontal distance is \frac{1}{3} of the run.

Apply the idea
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We can count the squares to determine the rise and the run. Then, we need to find \frac{1}{3} of each distance.

\frac{1}{3}\cdot \text{run}=\frac{1}{3}\cdot 6=2

\frac{1}{3}\cdot \text{rise}=\frac{1}{3}\cdot 12=4

From point A\left(-5,-4\right), moving right 2 and up 4, we get:

P=\left(-5+2,-4+4\right)=\left(-3,0\right)

The point that is \frac{1}{3} of the way from A to B is P\left(-3,0\right).

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Example 5

Find the point, P, that divides \overline{AB} in the ratio 3:5.

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Worked Solution
Create a strategy

To divide this segment into a ratio of 3:5, we need to divide the line into 8 equal parts with P being \dfrac{3}{8} of the way from A to B.

Apply the idea

We will use the same strategy as we did in the previous problem:

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\frac{3}{8}\cdot \text{run}=\frac{3}{8}\cdot 12=\frac{9}{2}

\frac{3}{8}\cdot \text{rise}=\frac{3}{8}\cdot -6=-\frac{9}{4}

From point A\left(-7,8\right), moving right \frac{9}{2} and down \frac{9}{4}, we get:

P=\left(-7+\frac{9}{2},8-\frac{9}{4}\right)=\left(-\frac{5}{2},\frac{23}{4}\right)

Reflect and check
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Graphing the result can help us check our answer visually.

P does appear to be \frac{3}{8} of the way from A to B.

Notice this is the same as having a ratio of 3:5 or being \frac{5}{8} of the way from B to A.

Idea summary

We can divide a segment in the ratio m:n by finding\left(\frac{m}{m + n}\right)\cdot\text{rise}\\ \left(\frac{m}{m + n}\right)\cdot\text{run}

and adding those values to the leftmost endpoint.

This is the same as finding the point that is \left(\dfrac{m}{m+n}\right) from the left endpoint to the right endpoint.

Outcomes

G.GPE.B.6

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

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