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SSAT Upper Level Math : Coordinate Geometry

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Example Questions

Example Question #2 : How To Find The Midpoint Of A Line Segment

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

Possible Answers:

(10, 7)

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

(5, 7)

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

Correct answer:

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

Explanation:

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

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

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Example Question #3 : 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?

Possible Answers:

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

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

(-1, 5)

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

Correct answer:

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

Explanation:

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

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

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

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Write the midpoint formula.

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

Substitute the values.

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

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Example Question #1 : How To Find The Endpoints Of A Line Segment

A line segment on the coordinate plane has midpoint $\left ( A + 5, B + 7\right )$ . One of its endpoints is $\left ( 10, 6 \right )$ . What is the -coordinate of the other endpoint, in terms of and/or ?

Possible Answers:

B + 8

2B + 20

2B + 1

B + 1

2B + 8

Correct answer:

2B + 8

Explanation:

Let be the -coordinate of the other endpoint. Since the -coordinate of the midpoint of the segment is the mean of those of the endpoints, we can set up an equation as follows:

$\frac{Y + 6 }{2} = B + 7$

$\frac{Y + 6 }{2} \cdot 2= \left ( B + 7 \right )\cdot 2$

Y + 6 = 2B + 14

Y + 6 -6 = 2B + 14 -6

Y = 2B + 8

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Example Question #2 : How To Find The Endpoints Of A Line Segment

A line segment on the coordinate plane has one endpoint at $\left ( 8, 10 \right )$ ; its midpoint is $\left ( A+B, A- B\right )$ . Which of the following gives the -coordinate of its other endpoint in terms of and ?

Possible Answers:

2 A + 2B+8

A + B- 16

2 A + 2B- 8

A + B- 8

A + B+8

Correct answer:

2 A + 2B- 8

Explanation:

To find the value of the -coordinate of the other endpoint, we will assign the variable . Then, since the -coordinate of the midpoint of the segment is the mean of those of its endpoints, the equation that we can set up is

$\frac{T + 8 }{2} = A + B$ .

We solve for :

$\frac{T + 8 }{2} \cdot 2 =\left ( A + B \right )\cdot 2$

T + 8 =2 A + 2B

T + 8 - 8 =2 A + 2B- 8

T =2 A + 2B- 8

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

One endpoint of a line segment on the coordinate plane is the point $\left (2\frac{4}{5}, -1\frac{1}{5} \right )$ ; the midpoint of the segment is the point $\left (6\frac{1}{2}, 4\frac{1}{2} \right )$ . Give the -coordinate of the other endpoint of the segment.

Possible Answers:

$-5\frac{1}{5}$

$4 \frac{3}{20}$

$10\frac{1}{5}$

$3\frac{7}{10}$

$9\frac{3}{10}$

Correct answer:

$10\frac{1}{5}$

Explanation:

In the part of the midpoint formula

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

set $x_{M}= 6\frac{1}{2}, x_{2}= 2\frac{4}{5}$ , and solve:

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

$6\frac{1}{2}= \frac{x_{1} + 2\frac{4}{5} }{2}$

$6\frac{1}{2} \cdot 2 = x_{1} + 2\frac{4}{5}$

$x_{1} + 2\frac{4}{5} = 13$

$x_{1} = 10\frac{1}{5}$

This is the correct -coordinate.

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

One endpoint of a line segment on the coordinate plane is the point $\left (8.32, -4.29 \right )$ ; the midpoint of the segment is the point (-1.89, 8.56) . Give the -coordinate of the other endpoint of the segment.

Possible Answers:

-12.1

3.215

2.135

21.41

-12.85

Correct answer:

21.41

Explanation:

Using the part of the midpoint formula

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

set $y_{M}=8.56, y_{2}= -4.29$ and solve:

$8.56= \frac{y_{1} -4.29 }{2}$

$y_{1} -4.29 = 17.12$

$y_{1} = 21.41$

The second endpoint is ( -12.1, 21.41) .

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

On the coordinate plane, is the midpoint of $\overline{AB}$ and is the midpoint of $\overline{AM}$ . has coordinates (288, 108) and has coordinates $\left ( 176, 300\right )$ .

Give the -coordinate of .

Possible Answers:

102

876

156

Correct answer:

876

Explanation:

First, find the -coordinate of . In the part of the midpoint formula

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

set $y_{M}=300, y_{2}= 108$ , and solve:

$300= \frac{y_{1} + 108}{2}$

$y_{1} + 108 = 600$

$y_{1} =492$

Now, find the -coordinate of similarly, setting

$y_{M}= 492,y_{2}= 108$

$492= \frac{y_{1} + 108}{2}$

$y_{1} + 108= 984$

$y_{1} = 876$

This is the correct response.

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Example Question #6 : How To Find The Endpoints Of A Line Segment

On the coordinate plane, is the midpoint of $\overline{AB}$ and is the midpoint of $\overline{AM}$ . has coordinates $\left (4\frac{1}{3},-12\frac{1}{2} \right )$ and has coordinates $\left ( -7 \frac{5}{6}, 6\frac{1}{3}\right )$ .

Give the -coordinate of .

Possible Answers:

$40\frac{5}{6}$

$62\frac{5}{6}$

$-44\frac{1}{3}$

$1\frac{1}{4}$

Correct answer:

$-44\frac{1}{3}$

Explanation:

First, find the -coordinate of . In the part of the midpoint formula

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

set $x_{M}= -7 \frac{5}{6}, x_{2}= 4\frac{1}{3}$ , and solve:

$-7 \frac{5}{6}= \frac{x_{1} + 4\frac{1}{3}}{2}$

$x_{1} + 4\frac{1}{3}= -7 \frac{5}{6} \cdot 2$

$x_{1} + 4\frac{1}{3}= -15 \frac{2}{3}$

$x_{1} = -15 \frac{2}{3} - 4\frac{1}{3}$

$x_{1} = -20$

Now, find the -coordinate of similarly, setting

$x_{M}= -20, x_{2}= 4\frac{1}{3}$

$-20 = \frac{x_{1} + 4\frac{1}{3}}{2}$

$x_{1} + 4\frac{1}{3}= -20 \cdot 2$

$x_{1} + 4\frac{1}{3}= -40$

$x_{1} = -40 - 4\frac{1}{3}$

$x_{1} = -44\frac{1}{3}$

This is the correct response.

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Example Question #2 : How To Find The Endpoints Of A Line Segment

Give the -coordinate of .

Possible Answers:

$-44\frac{1}{3}$

$62\frac{5}{6}$

$-50\frac{1}{6}$

$37\frac{5}{6}$

$-24\frac{2}{3}$

Correct answer:

$62\frac{5}{6}$

Explanation:

First, find the -coordinate of . In the part of the midpoint formula

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

set $y_{M}=6\frac{1}{3}, y_{2}= -12\frac{1}{2}$ , and solve:

$6\frac{1}{3}= \frac{y_{1} -12\frac{1}{2} }{2}$

$y_{1} -12\frac{1}{2}= 6\frac{1}{3} \cdot 2$

$y_{1} -12\frac{1}{2}=12 \frac{2}{3}$

$y_{1} =12 \frac{2}{3} + 12\frac{1}{2}$

$y_{1} = 25\frac{1}{6}$

Now, find the -coordinate of similarly, setting

$y_{M}=25\frac{1}{6}, y_{2}= -12\frac{1}{2}$

$25\frac{1}{6}= \frac{y_{1} -12\frac{1}{2} }{2}$

$y_{1} -12\frac{1}{2} = 25\frac{1}{6} \cdot 2$

$y_{1} -12\frac{1}{2} = 50\frac{1}{3}$

$y_{1} = 50\frac{1}{3} +12\frac{1}{2}$

$y_{1} = 62\frac{5}{6}$

This is the correct response.

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SSAT Upper Level Math : Coordinate Geometry

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #2 : How To Find The Midpoint Of A Line Segment

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

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

Example Question #1 : How To Find The Endpoints Of A Line Segment

Example Question #2 : How To Find The Endpoints Of A Line Segment

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

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

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

Example Question #6 : How To Find The Endpoints Of A Line Segment

Example Question #2 : How To Find The Endpoints Of A Line Segment

All SSAT Upper Level Math Resources

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