Midpoint Formula - SAT Math

Card 0 of 19

Question

The midpoint of line segment AB is (2, -5). If the coordinates of point A are (4, 4), what are the coordinates of B?

Answer

The fastest way to find the missing endpoint is to determine the distance from the known endpoint to the midpoint and then performing the same transformation on the midpoint. In this case, the x-coordinate moves from 4 to 2, or down by 2, so the new x-coordinate must be 2-2 = 0. The y-coordinate moves from 4 to -5, or down by 9, so the new y-coordinate must be -5-9 = -14.

An alternate solution would be to substitute (4,4) for (x1,y1) and (2,-5) for (x,y) into the midpoint formula:

x=(x1+x2)/2

y=(y1+y2)/2

Solving each equation for (x2,y2) yields the solution (0,-14).

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Question

Line segment AB has an endpoint, A, located at , and a midpoint at . What are the coordinates for point B of segment AB?

Answer

With an endpoint A located at (10,-1), and a midpoint at (10,0), we want to add the length from A to the midpoint onto the other side of the segment to find point B. The total length of the segment must be twice the distance from A to the midpoint.

A is located exactly one unit below the midpoint along the y-axis, for a total displacement of (0,1). To find point B, we add (10+0, 0+1), and get the coordinates for B: (10,1).

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Question

Point A is (5, 7). Point B is (x, y). The midpoint of AB is (17, –4). What is the value of B?

Answer

Point A is (5, 7). Point B is (x, y). The midpoint of AB is (17, –4). What is the value of B?

We need to use our generalized midpoint formula:

MP = ( (5 + x)/2, (7 + y)/2 )

Solve each separately:

(5 + x)/2 = 17 → 5 + x = 34 → x = 29

(7 + y)/2 = –4 → 7 + y = –8 → y = –15

Therefore, B is (29, –15).

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Question

Solve each problem and decide which is the best of the choices given.

What is the distance between the points and on a standard coordinate plane?

Answer

Make a triangle. The points are 8 units apart on the -axis, and units apart on the -axis. Then use the Pythagorean Theorem to find the distance of the hypotenuse, which ends up being $\sqrt{73}$ .

Another way to solve this problem is to use the distance formula,

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

Plugging in the two points we get,

$d=\sqrt{(1--7)^2+(6-3)^2}$

$d=\sqrt{(8)^2+(3)^2}$

$d=\sqrt{64+9}$

$d=\sqrt{73}$

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Question

Find the midpoint of the line segment with endpoints (1,3) and (5,7).

Answer

To solve, simply use the midpoint formula as outlined below.

Given,

Thus,

$midpoint=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)=\left(\frac{1+5}{2},\frac{3+7}{2}\right)=(3,5)$

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Question

A line segment has endpoints (0,4) and (5,6). What are the coordinates of the midpoint?

Answer

A line segment has endpoints (0,4) and (5,6). To find the midpoint, use the midpoint formula:

X: (x1+x2)/2 = (0+5)/2 = 2.5

Y: (y1+y2)/2 = (4+6)/2 = 5

The coordinates of the midpoint are (2.5,5).

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Question

Find the midpoint between (-3,7) and (5,-9)

Answer

You can find the midpoint of each coordinate by averaging them. In other words, add the two x coordinates together and divide by 2 and add the two y coordinates together and divide by 2.

x-midpoint = (-3 + 5)/2 = 2/2 = 1

y-midpoint = (7 + -9)/2 = -2/2 = -1

(1,-1)

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Question

Find the coordinates for the midpoint of the line segment that spans from (1, 1) to (11, 11).

Answer

The correct answer is (6, 6). The midpoint formula is ((x1 + x2)/2),((y1 + y2)/2) So 1 + 11 = 12, and 12/2 = 6 for both the x and y coordinates.

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Question

What is the midpoint between the points (–1, 2) and (3, –6)?

Answer

midpoint = ((x1 + x2)/2, (y1 + y2)/2)

= ((–1 + 3)/2, (2 – 6)/2)

= (2/2, –4/2)

= (1,–2)

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Question

$\overline{AB}$ has endpoints and .

What is the midpoint of $\overline{AB}$ ?

Answer

The midpoint is simply the point halfway between the x-coordinates and halfway between the y-coordinates:

Sum the x-coordinates and divide by 2:

$11\div 2=5.5$

Sum the y-coordinates and divide by 2:

$13\div 2=6.5$

Therefore the midpoint is (5.5, 6.5).

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Question

A line segment connects the points $(-1,4)$ and $(3,16)$ . What is the midpoint of this segment?

Answer

To solve this problem you will need to use the midpoint formula:

$midpoint = (\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2} )$

Plug in the given values for the endpoints of the segment: $(-1,4)$ and $(3,16)$ .

$midpoint = (\frac{-1+3}{2},\frac{4+16}{2} ) = (\frac{2}{2}, \frac{20}{2}) = (1, 10)$

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Question

What is the midpoint between and ?

Answer

The midpoint is the point halfway between the two endpoints, so sum up the coordinates and divide by 2:

$(\frac{1+5}{2},\frac{4+12}{2})=(\frac{6}{2},\frac{16}{2})=(3,8)$

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Question

What is the mid point of a linear line segment that spans from to ?

Answer

To find the midpoint of a line segment, use this formula:

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

Therefore, the midpoint of the given line segment is:

$\left ( \frac{-2+5}{2}\right ),\left ( \frac{4+14}{2} \right )$

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Question

Find the midpoint of a line segment with end points (1,3) and (11,3).

Answer

To solve, simply realize you are on a horizantal line, so you just need to find the distance betweent he two x coordants and find half way between them. Thus,

$\frac{10}{2}=5$

Thus, the midpoint is at (1+5,3) which is (6,3).

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Question

What is the midpoint of a line segment that begins at (0, -1) and ends at (4, 10)?

Answer

The midpoint of a line segment can be found by averaging the x-values and y-values of the given ordered pairs. In other words,

$M=(\frac{x_{1}+x_2}{2}, \frac{y_1+y_2}{2})$ .

Take the average of the given x- and y-values of our ordered pairs.

$M=(\frac{0+4}{2}, \frac{-1+10}{2})=(2, \frac{9}{2})$

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Question

Find the midpoint between the point and the center of the given circle.

Answer

Remember that the general equation for a circle with center and radius is .

With this in mind, the center of our circle is .

To find the midpoint between our two points, use the midpoint formula.

$M = (\frac{x_2+x_1}{2}, \frac{y_2+y_1}{2}) = (\frac{0.75+1.25}{2}, \frac{0.5+2.5}{2}) = (\frac{2}{2}, \frac{3}{2}) = (1, 1.5)$

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Question

The following coordinates represent the vertices of a box. Where does the center of the box lie?

Answer

To solve this, we need to choose two points that lie diagonally across from each other. Let's use and . Now, we can substitute these points into the Midpoint Formula.

$M = (\frac{x_2+x_1}{2}, \frac{y_2+y_1}{2}) = (\frac{9+1}{2}, \frac{9+1}{2}) = (\frac{10}{2}, \frac{10}{2}) = (5, 5)$

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Question

Give the coordinates of the midpoint, in terms of , of a segment on the coordinate plane whose endpoints are $\left (0.4 , N \right )$ and $\left ( N, -0.4 \right )$ .

Answer

The coordinates $(x_{M}, y_{M})$ of the midpoint of the line segment with endpoints $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ can be calculated using the formulas

$x_{M} =\frac{ x_{1} + x_{2} }{2}$

and

$y_{M} =\frac{ y_{1} + y_{2} }{2}$ .

Setting $x_{1} = 0.4$ and $x_{2} =N$ , and substituting:

$x_{M} =\frac{ 0.4 + N }{2} = \frac{ N }{2} + \frac{ 0.4 }{2} = 0.5 N + 0.2$

Setting $y_{1} =N$ and $y_{2} = -0.4$ , and substituting:

$x_{M} =\frac{N + (-0.4) }{2} = \frac{ N }{2} - \frac{ 0.4 }{2} = 0.5 N - 0.2$

The coordinates of the midpoint, in terms of , are

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Question

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

Answer

In order to find the midpoint of a line segment, you need to average the x and y values of the endpoints.

The midpoint formula is

$((x_{1}+x_{2})/2), ((y_{1}+y_{2})/2)$

After plugging in the values you get

for x

and for y

Therefore, the midpoint is at .

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