Midpoint Formula

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GED Math › Midpoint Formula

Questions 1 - 10

1

Find the midpoint of the line segment that connects the following points:

$\small (-5,-3)\ and\ (3,7)$

$\small (-1,2)$

$\small (-4,-5)$

$\small (4,5)$

$\small (1,2)$

Explanation

Use the midpoint formula:

$\small mid=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

$\small mid=(\frac{-5+3}{2},\frac{-3+7}{2})$

$\small mid=(\frac{-2}{2},\frac{4}{2})$

$\small mid=(-1,2)$

2

A line on a coordinate graph has endpoints at and . What is the midpoint of the line?

$(-3, \frac{7}{2})$

$(3, \frac{7}{2})$

$(-7, \frac{3}{2})$

$(3, -\frac{5}{2})$

Explanation

Recall how to find the midpoint of a line when given two endpoints:

$\text{Midpoint}=(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$

Plug in the given points to find the midpoint.

$\text{Midpoint}=(\frac{4+(-10)}{2}, \frac{5+2}{2})=(-3, \frac{7}{2})$

The midpoint of the line is located at $(-3, \frac{7}{2})$ .

3

Find the point midway between the following two points.

Explanation

Find the point midway between the following two points.

To find the midpoint, we need to essentially average our x and y values separately.

$x_m=\frac{x_1+x_2}{2}=\frac{54+99}{2}=\frac{153}{2}=76.5$

$y_m=\frac{y_1+y_2}{2}=\frac{45+32}{2}=\frac{77}{2}=38.5$

So, our answer is

4

What is the midpoint between the points and ?

Explanation

Recall that the general formula for the midpoint between two points is:

$(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Think of this like being the "average" of your two points.

Based on your data, you know that your midpoint could be calculated as follows:

$(\frac{-10-22}{2},\frac{4-4}{2})=(\frac{-32}{2},\frac{0}{2})$

This is the same as:

5

Find the midpoint of and .

$(-\frac{1}{2}, \frac{1}{2})$

$(-\frac{3}{2}, \frac{1}{2})$

$(-\frac{3}{2}, \frac{13}{2})$

Explanation

Write the formula for the midpoint.

$M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Substitute the points.

$M=(\frac{1+(-2)}{2},\frac{7+(-6)}{2}) = (\frac{-1}{2}, \frac{1}{2})$

The answer is: $(-\frac{1}{2}, \frac{1}{2})$

6

is the midpoint between and some other point. What is that point?

Explanation

Recall that the general formula for the midpoint between two points is:

$(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Think of this like being the "average" of your two points.

Now, you can write your data out as follows, as you know the midpoint value as well as one of the values for your end points:

$(0,5)=(\frac{84+x_2}{2},\frac{1+y_2}{2})$

To finish solving, think of it like two different equations:

$\frac{84+x_2}{2}=0$

and

$\frac{1+y_2}{2}=5$

Now, solve each for the respective values:

and

Therefore, your other point is or

7

Find the midpoint given the following points:

and

Explanation

To find the midpoint between two points, we will use the following formula:

$M = (\frac{x_1 +x_2}{2}), (\frac{y_1 +y_2}{2})$

where and are the given points.

So, given the points and , we can substitute. We get

$M = (\frac{9+1}{2}, \frac{1+2}{2})$

$M = (\frac{10}{2}, \frac{3}{2})$

8

What is the midpoint between and ?

$(\frac{3}{2}, -\frac{3}{2})$

$(\frac{3}{2}, -\frac{15}{2})$

$(\frac{1}{2}, -\frac{15}{2})$

$(-\frac{1}{2}, -\frac{15}{2})$

$(-\frac{3}{2}, -\frac{3}{2})$

Explanation

Write the formula to find the midpoint.

$M = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$

Substitute the points into the equation.

$M = (\frac{2+1}{2}, \frac{6+(-9)}{2})$

The midpoint is located at: $(\frac{3}{2}, -\frac{3}{2})$

The answer is: $(\frac{3}{2}, -\frac{3}{2})$

9

The midpoint of a line with endpoints at and is . Find the value of .

Explanation

Recall how to find the midpoint of a line:

$\text{Midpoint}=(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$

Since we are only worried about the -coordinate, we can write the following equation:

$\frac{5+y}{2}=12$

Solve for .

10

You are given points and . is the midpoint of $\overline{AD}$ , is the midpoint of $\overline{AC}$ , and is the midpoint of $\overline{BD}$ . Give the coordinates of .

$\left ( 7.625, 2\right )$

Explanation

Repeated application of the midpoint formula, $\left (\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2} \right )$ , yields the following:

is the point and is the point $\left (1, 9 \right )$ . is the midpoint of $\overline{AD}$ , so has coordinates

$\left (\frac{8+1}{2}, \frac{1+9}{2} \right )$ , or $\left ( 4.5, 5\right )$ .

is the midpoint of $\overline{AC}$ , so has coordinates

$\left (\frac{8+4.5}{2}, \frac{1+5}{2} \right )$ , or .

is the midpoint of $\overline{BD}$ , so has coordinates

$\left (\frac{6.25+1}{2}, \frac{3+9}{2} \right )$ , or .

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