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9. Connecting Algebra & Geometry with Coordinates
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United States of AmericaTennessee
Geometry

9.03 Midpoints

Lesson
Practice
Lesson

Concept summary

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

Endpoint

The point at the end of a segment or the starting point of a ray. A line segment will have two endpoints.

A line segment with a point at either end. One of the points is highlighted.
Midpoint

The point that divides a line segment into two congruent line segments.

A line segment with endpoints A and C with a point B directly in the middle labeled Midpoint

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

For endpoints A and B, we may also see this formula written as:M=\frac{1}{2}A+\dfrac{1}{2}B = \frac{A + B}{2}

Notice that this looks the same as taking the mean (average) of two values. To find a mean, we sum the values together and divide by the number of values.

Another strategy for finding the midpoint is to use similar triangles:

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y

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

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We can create a similar triangle that is \dfrac{1}{2} the size of the original triangle.

\dfrac{1}{2}\text{run}=\dfrac{1}{2}\cdot 5=2.5

\dfrac{1}{2}\text{rise}=\dfrac{1}{2}\cdot 10=5

From point A\left(2,1\right), moving right 2.5 and up 5, we get:

\left(2+2.5,1+5\right)=\left(4.5,6\right)

Worked examples

Example 1

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

-7
-6
-5
-4
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-4
-3
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-1
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Find the midpoint, M, of \overline{AB}.

Solution

\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+1}{2}, \dfrac{-4+8}{2} \right)Substitute in coordinates of endpoints
\displaystyle =\displaystyle \left(\dfrac{-4}{2}, \dfrac{4}{2} \right)Simplify numerators
\displaystyle =\displaystyle \left(-2, 2 \right)Simplify fractions

Reflection

When finding the midpoint of a line segment, the order of the endpoints doesn't matter, since M is the same distance from both endpoints.

Example 2

If the midpoints of A \left(a,b\right) and B \left(1,4\right) is M\left(9,7\right):

a

Find the value of a.

Approach

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.

Solution

\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{a+1}{2}Substitute in the x-coordinates
\displaystyle 18\displaystyle =\displaystyle a+1Multiply 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.

Approach

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

Solution

\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{b+4}{2}Substitute in the y-coordinates
\displaystyle 14\displaystyle =\displaystyle b+4Multiply both sides by 2
\displaystyle 10\displaystyle =\displaystyle bSubtract 4 from both sides
\displaystyle b\displaystyle =\displaystyle 10Symmetric property of equality

Reflection

This means that the coordinates of A are \left(17,10\right).

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)

Approach

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

Solution

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 in given values
\displaystyle M\displaystyle =\displaystyle \left(36.555, -101.72\right)Evaluate

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 between Dalhart and Liberal.

Reflection

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.

Outcomes

G.N.Q.A.1

Use units as a way to understand real-world problems.*

G.N.Q.A.1.A

Use appropriate quantities in formulas, converting units as necessary.

G.GPE.A.1

Use coordinates to justify geometric relationships algebraically and to solve problems.

G.MP1

Make sense of problems and persevere in solving them.

G.MP2

Reason abstractly and quantitatively.

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP4

Model with mathematics.

G.MP5

Use appropriate tools strategically.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

G.MP8

Look for and express regularity in repeated reasoning.

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