Coordinate Geometry - GED Math

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Question

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

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

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Answer

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}$)$

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

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Question

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 .

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Answer

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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Question

You are given and . is the midpoint of $\overline{AC}$ ; is the midpoint of $\overline{AD}$ . What are the coordinates of ?

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Answer

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

Since is the midpoint of $\overline{AC}$ , substitute the coordinates of for and , set equal to the coordinates of , and solve as follows:

$7 = $$\frac{3+x_{2}$$}{2}$

$14 $=3+x_{2}$$

$-1= $$\frac{7+y_{2}$$}{2}$

$-2= $7+y_{2}$$

$-9= $y_{2}$$

is the point .

We can find the coordinates of similarly using those of and :

$11= $$\frac{3+x_{2}$$}{2}$

$22 = $3+x_{2}$$

$19= $x_{2}$$

$-9= $$\frac{7+y_{2}$$}{2}$

$-18= $7+y_{2}$$

$-25= $y_{2}$$

is the point

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Question

What is the midpoint between and ?

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Answer

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}$)$

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Question

Find the midpoint of and .

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Answer

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

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Question

What is the midpoint between and ?

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Answer

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{4-7}{2}$,$\frac{5+10}{2}$)=($\frac{-3}{2}$,$\frac{15}{2}$)$

This is the same as:

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Question

What is the midpoint between the points and ?

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Answer

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:

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Question

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

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Answer

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

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Question

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

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Answer

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:

$(-10,-22)=($\frac{-2+x_2}{2}$,$\frac{-7+y_2}{2}$)$

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

$$\frac{-2+x_2}{2}$=-10$

and

$$\frac{-7+y_2}{2}$=-22$

Now, solve each for the respective values:

and

Therefore, your other point is or

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Question

Find the midpoint of the following points:

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Answer

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 into the formula. We get

$M = ($\frac{8+10}{2}$, $\frac{2+4}{2}$)$

$M = ($\frac{18}{2}$, $\frac{6}{2}$)$

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Question

Find the midpoint given the following points:

and

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Answer

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

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Question

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

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Answer

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 .

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Question

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

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Answer

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, $\frac{7}{2}$)$ .

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Question

Find the point halfway between the two following points:

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Answer

Find the point halfway between the two following points:

We need to use the midpoint formula. To do so, we are basically averaging out our x and y values separately.

So, plug in our values to solve:

So, our answer is (15,49)

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Question

If is the midpoint of a line that has endpoints at and , find the value of .

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Answer

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

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

Since the y-coordinate of the midpoint is , we can write the following equation:

$14=\frac{-2+y}{2}$$

Solve for .

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Question

Find the point midway between the following two points.

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Answer

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

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Question

The midpoint of a line segment on the coordinate plane is located at ; one endpoint is located at the origin. Where is the other endpoint?

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Answer

The midpoint of a line segment with its endpoints at and is at the point , where

and .

The midpoint and one endpoint are given, so first, find the -coordinate of the second endpoint by setting and , and solving for :

$3 = $\frac{ $x_{1}$$ +0}{2}$

$3 = $\frac{ $x_{1}$$ }{2}$

Isolate by multiplying both sides by 2:

$3 \cdot 2 = $\frac{ $x_{1}$$ }{2} \cdot 2$

$6 = $x_{1}$$

Find the -coordinate of the second endpoint similarly, setting and :

$9 = $\frac{ $y_{1}$$ +0}{2}$

$9 = $\frac{ $y_{1}$$ }{2}$

$9 \cdot 2 = $\frac{ $y_{1}$$ }{2} \cdot 2$

$18 = $y_{1}$$

The second endpoint of the segment is at .

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Question

The midpoint of a line that passes through the points and is . Find the coordinates of the point .

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Answer

Recall how to find the midpoint of a line:

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

Start by finding the x-coordinate.

$$\frac{5+x}{2}$=4$

Next, find the y-coordinate.

$$\frac{1+y}{2}$=-2$

The coordinates for the second endpoint is

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Question

If the midpoint of a line with one endpoint at is , what are the coordinates for the other endpoint?

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Answer

Recall how to find the midpoint of a line segment:

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

Start by finding the x-coordinate of the other endpoint.

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

Next, find the y-coordinate of the other endpoint.

$-10=\frac{1+y}{2}$$

The other endpoint must be .

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Question

The two endpoints of a line segment are and . Find the midpoint of the line segment.

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Answer

To find the midpoint of a line, you basically have to average the two values and average the two values. In other words, you have to add the two values and then divide by , and add the two values and then divide by . The standard equation for the midpoint formula is .

The two values are and , and , so the coordinate of your midpoint is . The two values are and , and , so the coordinate of your midpoint is .

Therefore, the midpoint is .

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