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Intermediate Geometry : Midpoint Formula

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Example Question #1 :Coordinate Geometry

What is the midpoint between (-1,3) and (7,-5) ?

Possible Answers:

(4,-4)

(2,2)

(3,0)

(3,-1)

(4,-1)

Correct answer:

(3,-1)

解释:

The midpoint is given by taking the mean, or average, of theandcoordinates separately.

Let $P_{1}=\left ( 7,5 \right )$ and $P_{2} = \left (-1,3 \right )$

So the midpoint formula becomes $\frac{x_{1}\ +\ x_{2}}{2}= \frac{7-1}{2}=3$ and $\frac{y_{1}+y_{2}}{2}= \frac{-5 +3}{2}=-1$

So the midpoint is (3,-1)

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Example Question #1 :Coordinate Geometry

What is the midpoint of a line segment connecting the points $\left ( -5,6 \right )$ and $\left ( -3,2 \right )\ ?$

Possible Answers:

$\left ( -1,4 \right )$

$\left ( -4,4 \right )$

$\left ( -4,2 \right )$

$\left ( -1,2 \right )$

Correct answer:

$\left ( -4,4 \right )$

解释:

Use the midpoint formula:

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

$(\frac{-5+(-3)}{2}, \frac{6+2}{2})=(\frac{-8}{2}, \frac{8}{2})=(-4,4)$

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Example Question #3 :How To Find The Midpoint Of A Line Segment

Given two points $\left ( -3,-3 \right )$ and $\left ( -1,1 \right )$ and a line segment that connects the two.

What is the midpoint of the line segment? Addtionally, what is the length of the line segment?

Possible Answers:

Midpoint: $\left ( -2,-1 \right )$

Length: $\sqrt5$

Midpoint: $\left ( -1,-2 \right )$

Length: $2\sqrt5$

Midpoint: $\left ( -2,-1 \right )$

Length: $2\sqrt{5}$

Midpoint: $\left ( -1,-2 \right )$

Length:

Midpoint: $\left ( -1,-1 \right )$

Length: $4\sqrt3$

Correct answer:

Midpoint: $\left ( -2,-1 \right )$

Length: $2\sqrt{5}$

解释:

The midpoint formula is as follows:

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

This makes sense; it is as simple as the average of theandcomponents of each point.

For this problem, $x_1=-3,x_2=-1,y_1=-3,and\: y_2=1$ .

So the midpoint is $\left ( -2,-1 \right )$

The distance formula is as follows:

$distance=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

The order you put the first and second components of each point does NOT matter, when a positive or negative number is squared, it will always come out positive.

$(2)^2=4\; and\; (-4)^2=16$

So the distance is $\sqrt{(-3+1)^2+(-3-1)^2} =\sqrt{4+16}=\sqrt{20}=2\sqrt{5}$

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Example Question #4 :How To Find The Midpoint Of A Line Segment

Find the midpoint of a line segment going from (-2,-9) to (-6,-1) .

Possible Answers:

(2,4)

(3,5)

(-4,-5)

(-4,4)

(-1,3)

Correct answer:

(-4,-5)

解释:

To find the midpoint of a line segment, you must find the mean of the x values and the mean of the y values.

Our x values areandto find their mean, we do (-2+-6)/2=-4 .

Our y values areand, so our mean is (-9+-1)/2=-5 . Therefore, our midpoint must be (-4,-5) .

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Example Question #5 :How To Find The Midpoint Of A Line Segment

Find the coordinate of the midpoint of the line segment connecting the pair of points

(2, 2) and (4, 12) .

Possible Answers:

(2, 7)

(1, 5)

(3, 7)

Cannot be determined

Correct answer:

(3, 7)

解释:

The coordinate of the midpoint of the line segment connecting a pair of points is

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

So for the pair of points (x_1,y_1)=(2,2) and (x_2,y_2)=(4,12) ,

we get:

$\left(\frac{2+4}{2}, \frac{2+12}{2}\right)=(3, 7)$

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Example Question #1 :Lines

Find the coordinate of the midpoint of the line segment connecting the pair of points

(0, 0) and (8, 0) .

Possible Answers:

(-4, 0)

(4, 0)

(0, 4))

Cannot be determined

Correct answer:

(4, 0)

解释:

The coordinate of the midpoint of the line segment connecting a pair of points is

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

So for the pair of points (x_1,y_1)=(0, 0) and (x_2,y_2)=(8, 0)

we get,

$\left(\frac{0+8}{2}, \frac{0+0}{2}\right)=(4, 0)$

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Example Question #7 :How To Find The Midpoint Of A Line Segment

A line extends from the origin to $\small (14, -2)$ . What is its midpoint?

Possible Answers:

$\small (7,-1)$

$\small (7, 1 )$

$\small (28, -4)$

$\small (1,-7)$

$\small (-7, -1)$

Correct answer:

$\small (7,-1)$

解释:

The origin is the point $\small (0,0)$ . We can use the regular midpoint formula with $\small (0,0)$ as $\small (x_{1}, y_{1})$ and the given point $\small (14, -2)$ as $\small (x_{2}, y_{2})$ :

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

$\small (\frac{0+14}{2} , \frac{0+-2}{2})$

$\small (\frac{14}{2}, \frac{-2}{2}) = (7, -1)$

Note that since the first point is the origin, we're averaging these points with 0, or in other words just dividing by 2.

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Example Question #8 :How To Find The Midpoint Of A Line Segment

There is a line segment between the two points (5, 10) and (3, 6). What formula will give you the correct mid-point of the line segment?

Possible Answers:

$\Big( {5-2} \; ,{10-2} \Big )$

$\Big( \frac{5-3}{2} \; , \frac{10-6}{2} \Big )$

$\Big( \frac{5+10}{2} \; , \frac{3+6}{2} \Big )$

$\Big( \frac{5+3}{2} \; , \frac{10+6}{2} \Big )$

$\Big( \frac{5-10}{2} \; , \frac{3-6}{2} \Big )$

Correct answer:

$\Big( \frac{5+3}{2} \; , \frac{10+6}{2} \Big )$

解释:

To find the mid-point given two points on the coordinate plane you are basically finding the average of the x-values and the average of the y-values. To do this follow the formula below:

$\Big( \frac{x_{1}+ x_{2}}{2} \; , \frac{y_{1}+y_{2}}{2} \Big )$

With our points this gives:

$\Big( \frac{5+3}{2} \; , \frac{10+6}{2} \Big )$

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Example Question #9 :How To Find The Midpoint Of A Line Segment

A line has endpoints at (4,5) and (-6,8). Where is its midpoint?

Possible Answers:

$\left(-1,\frac{13}{2}\right)$

$\left(\frac{13}{2},-1\right)$

$\left(\frac{-13}{2},1\right)$

$\left(1,\frac{13}{2}\right)$

$\left(\frac{-13}{2},-1\right)$

Correct answer:

$\left(-1,\frac{13}{2}\right)$

解释:

The midpoint is the point that is equidistant from each endpoint point.

We find it using a formula that computes the average of the x and y coordinates.

$\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)$

$\left(\frac{4+(-6)}{2},\frac{5+8}{2}\right)$

$\left(\frac{-2}{2},\frac{13}{2}\right)$

$\mathbf{\left(-1,\frac{13}{2}\right)}$

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Example Question #10 :How To Find The Midpoint Of A Line Segment

A line segment has the endpoints (2, 4) and (1, 8) . What is the midpoint of this line segment?

Possible Answers:

$(\frac{3}{2},6)$

$(\frac{1}{2},12)$

(6, 3)

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

Correct answer:

$(\frac{3}{2},6)$

解释:

Recall the formula for finding a midpoint of a line segment:

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

The coordinates of the midpoint is just the average of the x-coordinates and the average of the y-coordinates.

Plug in the given points to find the midpoint of the line segment.

$\text{Midpoint}=(\frac{2+1}{2}, \frac{4+8}{2})=(\frac{3}{2}, 6)$

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Intermediate Geometry : Midpoint Formula

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 :Coordinate Geometry

Example Question #1 :Coordinate Geometry

Example Question #3 :How To Find The Midpoint Of A Line Segment

Example Question #4 :How To Find The Midpoint Of A Line Segment

Example Question #5 :How To Find The Midpoint Of A Line Segment

Example Question #1 :Lines

Example Question #7 :How To Find The Midpoint Of A Line Segment

Example Question #8 :How To Find The Midpoint Of A Line Segment

Example Question #9 :How To Find The Midpoint Of A Line Segment

Example Question #10 :How To Find The Midpoint Of A Line Segment

All Intermediate Geometry Resources

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