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Intermediate Geometry : Coordinate Geometry

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Example Questions

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Example Question #27 : How To Find The Endpoints Of A Line Segment

A line segment has an endpoint at (8, -2) and midpoint at (1, -1) . Find the coordinates of the other endpoint.

Possible Answers:

(-5, 0)

(2, 0)

(2, 10)

(-6, 0)

Correct answer:

(-6, 0)

Explanation:

Recall how to find the midpoint of a line segment:

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

where $(x_1, y_1)\text{ and}(y_1, y_2)$ are the endpoints.

Let's first focus on the coordinate of the other endpoint. Using the information given by the question, we can write the following equation:

$\frac{8+x_2}{2}=1$

Solve for x_2 .

8+x_2=2

x_2=-6

Next, find the coordinate of the other endpoint. Using the information given by the question, we can write the following equation:

$\frac{-2+y_2}{2}=-1$

-2+y_2=-2

y_2=0

The second endpoint must be at (-6, 0) .

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Example Question #28 : How To Find The Endpoints Of A Line Segment

A line segment has an endpoint at (9, -3) and midpoint at (12, -8) . Find the coordinates of the other endpoint.

Possible Answers:

(15, -13)

(16, -21)

(19, -4)

(17, -18)

Correct answer:

(15, -13)

Explanation:

Recall how to find the midpoint of a line segment:

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

where $(x_1, y_1)\text{ and}(y_1, y_2)$ are the endpoints.

Let's first focus on the coordinate of the other endpoint. Using the information given by the question, we can write the following equation:

$\frac{9+x_2}{2}=12$

Solve for x_2 .

9+x_2=24

x_2=15

Next, find the coordinate of the other endpoint. Using the information given by the question, we can write the following equation:

$\frac{-3+y_2}{2}=-8$

-3+y_2=-16

y_2=-13

The second endpoint must be at (15, -13) .

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Example Question #29 : How To Find The Endpoints Of A Line Segment

A line segment has an endpoint at (17, 2) and midpoint at (6, 8) . Find the coordinates of the other endpoint.

Possible Answers:

(16, 12)

(18, 20)

(18, 17)

(15, 14)

Correct answer:

(15, 14)

Explanation:

Recall how to find the midpoint of a line segment:

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

where $(x_1, y_1)\text{ and}(y_1, y_2)$ are the endpoints.

Let's first focus on the coordinate of the other endpoint. Using the information given by the question, we can write the following equation:

$\frac{17+x_2}{2}=6$

Solve for x_2 .

17+x_2=32

x_2=15

Next, find the coordinate of the other endpoint. Using the information given by the question, we can write the following equation:

$\frac{2+y_2}{2}=8$

2+y_2=16

y_2=14

The second endpoint must be at (15, 14) .

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Example Question #30 : How To Find The Endpoints Of A Line Segment

A line segment has an endpoint at (-2, 4) and a midpoint at (12, 2) . Find the other endpoint.

Possible Answers:

(28, -2)

(-14, 22)

(26, 0)

(30, 4)

Correct answer:

(26, 0)

Explanation:

Recall how to find the midpoint of a line segment:

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

where $(x_1, y_1)\text{ and}(y_1, y_2)$ are the endpoints.

Let's first focus on the coordinate of the other endpoint. Using the information given by the question, we can write the following equation:

$\frac{-2+x_2}{2}=12$

Solve for x_2 .

-2+x_2=24

x_2=26

Next, find the coordinate of the other endpoint. Using the information given by the question, we can write the following equation:

$\frac{4+y_2}{2}=2$

4+y_2=4

y_2=0

The second endpoint must be at (26, 0) .

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Example Question #31 : How To Find The Endpoints Of A Line Segment

A line segment on the coordinate plane has an endpoint at (3.8, -1.7) ; its midpoint is at (9.2, 5.5) .

True or false: Its other endpoint is located at (14.6,12.7 ) .

Possible Answers:

False

True

Correct answer:

True

Explanation:

The midpoint of a line segment with endpoints $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is located at $\left ( \frac{x_{1}+x_{2}}{2} , \frac{y_{1}+ y_{2}}{2} \right )$ .

Therefore, set

$\frac{x_{1}+x_{2}}{2} = 9.2$ and $\frac{y_{1}+ y_{2}}{2} = 5.5$

In the first equation, set $x_{2} = 3.8$ and solve for $x_{1}$ :

$\frac{x_{1}+3.8}{2} = 9.2$

Multiply both sides by 2:

$\frac{x_{1}+3.8}{2} \cdot 2 = 9.2 \cdot 2$

$x_{1}+3.8= 18.4$

Subtract 3.8 from both sides:

$x_{1}+3.8- 3.8 = 18.4 - 3.8$

$x_{1} = 14.6$

In the second equation, set $y_{2} = -1.7$ and solve for $x_{2}$ :

$\frac{y_{1}+ (-1.7)}{2} = 5.5$

Multiply both sides by 2:

$\frac{y_{1} -1.7}{2} \cdot 2 = 5.5 \cdot 2$

$y_{1} -1.7 = 11$

Add 1.7 to both sides:

$y_{1} -1.7+ 1.7 = 11 + 1.7$

$y_{1} = 12.7$

The other endpoint is indeed at (14.6,12.7 ) , so the statement is true.

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Example Question #1 : Distance Formula

A line segment begins from the origin and is 10 units long, which of the following points could NOT be an endpoint for the line segment?

Possible Answers:

$(\sqrt{15},-\sqrt{85})$

(5,9)

(6,8)

(-6,8)

$(7,\sqrt{51})$

Correct answer:

(5,9)

Explanation:

By the distance formula, the sum of the squares of each point must add up to 10 squared. The only point that doesn't fufill this requirement is (5,9)

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Example Question #2 : Distance Formula

A line segment is drawn starting from the origin and terminating at the point (15,8) . What is the length of the line segment?

Possible Answers:

$\sqrt{264}$

$\sqrt{285}$

Correct answer:

Explanation:

Using the distance formula, $\sqrt{8^{2}+15^{2}} = 17$

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Example Question #1 : How To Find The Length Of A Line With Distance Formula

What is the distance between (-2,-3) and (6,12) ?

Possible Answers:

Correct answer:

Explanation:

In general, the distance formula is given by: $d=\sqrt{(x_{2}-x_{1})^{2}+{(y_{2}-y_{1})^{2}}}$ and is based on the Pythagorean Theorem.

Let $P_{1}=(-2,-3)$ and $P_{2}= (6,12)$

So the equation to soolve becomes $d=\sqrt{(6-(-2))^{2}+{(12-(3))^{2}}}$ or d=17

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Example Question #1 : Distance Formula

If we graph the equation y=-2x+4 what is the distance from the y-intercept to the x-intercept?

Possible Answers:

$2\sqrt{5}$

$2\sqrt{6}$

$\sqrt{5}$

$\sqrt{6}$

Correct answer:

$2\sqrt{5}$

Explanation:

First, you must figure out where the x and y intercepts lie. To do this we begin by plugging in y=0 to our equation, giving us -2x+4=0 . Thus x=2 . So our x-intercept is the point (2,0) . We then plug in x=0 , giving us y=4 , so we know our y-intercept is the point (0,4) . We then use the distance formula $d=\sqrt{(x_{1}-x_{2})^2+(y_{1}-y_{2})^2}$ and plug in our points, giving us $d=\sqrt{(2-0)^2+(0-4)^2}=\sqrt{20}=2\sqrt{5}$

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Example Question #1 : Distance Formula

Find the distance of the line connecting the pair of points

(0,0) and (5, 12) .

Possible Answers:

Correct answer:

Explanation:

By the distance formula

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

where (x_1,y_1)=(0,0) and (x_2,y_2)=(5,12)

we have

$\sqrt{(5-0)^2+(12-0)^2 }=\sqrt{(5)^2+(12)^2}= \sqrt{25+144} = \sqrt{169} = 13$

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Intermediate Geometry : Coordinate Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #27 : How To Find The Endpoints Of A Line Segment

Example Question #28 : How To Find The Endpoints Of A Line Segment

Example Question #29 : How To Find The Endpoints Of A Line Segment

Example Question #30 : How To Find The Endpoints Of A Line Segment

Example Question #31 : How To Find The Endpoints Of A Line Segment

Example Question #1 : Distance Formula

Example Question #2 : Distance Formula

Example Question #1 : How To Find The Length Of A Line With Distance Formula

Example Question #1 : Distance Formula

Example Question #1 : Distance Formula

All Intermediate Geometry Resources

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