SSAT Upper Level Quantitative - How to find the midpoint of a line segment

A line segment has endpoints $(-2, 0) \text{ and }(1, 5)$ . What is the midpoint of this line segment?

$\left(-\frac{1}{2}, \frac{5}{2}\right)$

$\left(-\frac{1}{2}, 5\right)$

$\left(5, -\frac{1}{2}\right)$

Explanation

To find the coordinates of the midpoint of a line, take the averages of the x and y coordinates of the endpoints.

$\text{Midpoint}=\left(\frac{-2+1}{2}, \frac{5+0}{2}\right)=\left(-\frac{1}{2}, \frac{5}{2}\right)$

What is the midpoint of a line segment with endpoints and ?

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

$(\frac{13}{2}, 0)$

Explanation

To find the midpoint of a line, just take the average of the -coordinates and the average of the -coordinates.

$\text{Midpoint}=(\frac{12+(-2)}{2}, \frac{2+1}{2})=(\frac{10}{2}, \frac{3}{2})=(5, \frac{3}{2})$

What is the midpoint of a line segments with endpoints and ?

Explanation

To find the midpoint of a line segment, take the average of the -coordinates, then take the average of the -coordinates.

$\text{Midpoint}=(\frac{2a+4a}{2}, \frac{-b+5b}{2})$

$\text{Midpoint}=(\frac{6a}{2}, \frac{4b}{2})=(3a, 2b)$

A line has the endpoints $(8,8) \text{ and }(2, -1)$ . What is its midpoint?

$\left(5, \frac{7}{2}\right)$

$\left(\frac{7}{2}, 5\right)$

Explanation

The coordinates of the midpoint of a line are just the averages of the coordinates of the endpoints.

$\text{Midpoint}=\left(\frac{8+2}{2}, \frac{8-1}{2}\right)=\left(5, \frac{7}{2}\right)$

A line segment has the endpoints and . What is the midpoint of the line?

Explanation

To find the midpoint of a line, take the averages of the x-coordinates and the average of the y-coordinates.

$\text{Midpoint}=(\frac{6+(-2)}{2}, \frac{10+(-4)}{2})=(\frac{4}{2}, \frac{6}{2})=(2,3)$

What is the midpoint of a line if the endpoints of the line are: $(6,9) \textup{ and }(3,-2)$ ?

$\left(\frac{9}{2},\frac{7}{2}\right)$

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

$\left(\frac{3}{2},\frac{11}{2}\right)$

$\left(-\frac{3}{2},-\frac{11}{2}\right)$

$\left(\frac{7}{2},\frac{9}{2}\right)$

Explanation

Write the midpoint formula.

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

Substitute the values.

$M=\left(\frac{6+3}{2},\frac{9-2}{2}\right)= \left(\frac{9}{2},\frac{7}{2}\right)$

A line segment on the coordinate plane has endpoints and . In terms of and , as applicable, give the -coordinate of its midpoint.

$\frac{a + b + 3}{2}$

$\frac{a - b - 4}{2}$

$\frac{a - b - 13}{2}$

Explanation

The -coordinate of the midpoint of a line segment is the mean of the -coordinates of its endpoints. Therefore, the -coordinate is

$x= \frac{(a-5)+(b+8)}{2} = \frac{a+b-5+8}{2} = \frac{a+b+3}{2}$ .

A line segment on the coordinate plane has endpoints and . In terms of and , as applicable, give the -coordinate of its midpoint.

$\frac{a - b - 4}{2}$

$\frac{a + b - 4}{2}$

$\frac{a + b + 3}{2}$

Explanation

The -coordinate of the midpoint of a line segment is the mean of the -coordinates of its endpoints. Therefore, the -coordinate is

$y= \frac{(a-4)+(-b)}{2} = \frac{a-b-4}{2}$ .

How to find the midpoint of a line segment

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