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ACT Math : How to find the endpoints of a line segment

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Example Questions

Example Question #1 : Midpoint Formula

In the standard (x,y) coordinate plane, the midpoint of line XY is (12, –3) and point X is located at (3, 4). What are the coordinates of point Y?

Possible Answers:

(9, 7)

(9, –7)

(21, –10)

(–4, 11)

(7.5, 0.5)

Correct answer:

(21, –10)

Explanation:

To get from the midpoint of (12, –3) to point (3,4), we travel –9 units in the x-direction and 7 units in the y-direction. To find the other point, we travel the same magnitude in the opposite direction from the midpoint, 9 units in the x-direction and –7 units in the y-direction to point (21, –10).

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Example Question #1 : Midpoint Formula

The midpoint of a line segment is (4, -1) . If one endpoint of the line segment is (-4, 3) , what is the other endpoint?

Possible Answers:

(-12,7)

(-12,-5)

(12,-5)

(12,7)

(8,6)

Correct answer:

(12,-5)

Explanation:

The midpoint formula can be used to solve this problem, where the midpoint is the average of the two coordinates.

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

We are given the midpoint and one endpoint. Plug these values into the formula.

$(4,-1)=(\frac{-4+x}{2},\frac{3+y}{2})$

$4=\frac{-4+x}{2}\ \text{and}\ -1=\frac{3+y}{2}$

Solve for the variables to find the coordinates of the second endpoint.

$8=-4+x\ \text{and}\ -2=3+y$

$12=x\ \text{and}\ -5=y$

The final coordinates of the other endpoint are (12,-5) .

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Example Question #2 : Midpoint Formula

Suppose the midpoint of a line segment is (4,12). What are the endpoints of the segment?

Possible Answers:

(2,12),(6,16)

(8,2),(6,16)

(2,8),(6,16)

(12,3),(3,12)

(3,12), (4,7)

Correct answer:

(2,8),(6,16)

Explanation:

The midpoint of a line segment is found using the formula $\left ( \frac{x_1+x_2 }{2}, \frac{y_1+y_2}{2}\right )$ .

The midpoint is given as (4,12). Going through the answer choices, only the points (2,8) and (6,16) yield the correct midpoint of (4,12) .

$\left ( \frac{2+6}{2}, \frac{8+16}{2} \right )=(4,12)$

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Example Question #3 : Midpoint Formula

What is the midpoint of the segment of

2y = 3x + 22

between x = 4 and x = 10 ?

Possible Answers:

(5.5,21.5)

(3,4.5)

(7,21.5)

(3,22.5)

(10,26)

Correct answer:

(7,21.5)

Explanation:

What is the midpoint of the segment of

2y = 3x + 22

between x = 4 and x = 10 ?

To find this midpoint, you must first calculate the two end points. Thus, substitute in for :

2y = 3* 4+22

2y = 34

y=17

Then, for x = 10 :

2y = 3 * 10 +22

2y = 52

y=26

Thus, the two points in question are:

(4,17) and (10,26)

The midpoint of two points is:

$(\frac{x_1+x_0}{2},\frac{y_1+y_0}{2})$

Thus, for our data, this is:

$(\frac{4+10}{2},\frac{17+26}{2})$

(7,21.5)

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Example Question #3 : Midpoint Formula

If (-10,-3) is the midpoint of (-20,15) and another point, what is that other point?

Possible Answers:

(-15,6)

(4,-18)

(2,-12)

(0,-21)

(-30,12)

Correct answer:

(0,-21)

Explanation:

Recall that the midpoint's and values are the average of the and values of the two points in question. Thus, if we call the other point (x,y) , we know that:

$-10 = \frac{-20+x}{2}$ and $-3 = \frac{15+y}{2}$

Solve each equation accordingly:

For $-10 = \frac{-20+x}{2}$ , multiply both sides by :

-20=-20+x

Thus, x=0

The same goes for the other equation:

-6=15+y , so y=-21

Thus, our point is (0,-21)

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Example Question #6 : Midpoint Formula

If (15,40) is the midpoint of (5,10) and another point, what is that other point?

Possible Answers:

(25,70)

(15,50)

(10,25)

(20,50)

(10,-5)

Correct answer:

(25,70)

Explanation:

If (15,40) is the midpoint of (5,10) and another point, what is that other point?

Recall that the midpoint's and values are the average of the and values of the two points in question. Thus, if we call the other point (x,y) , we know that:

$15 = \frac{5+x}{2}$ and $40 = \frac{10+y}{2}$

Solve each equation accordingly:

For $15 = \frac{5+x}{2}$ , multiply both sides by and then subtract from both sides:

30=5+x

Thus, x=25

For $40 = \frac{10+y}{2}$ , multiply both sides by 2 and then subtract 10 from both sides:

Thus, y=70

Thus, our point is (25,70)

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ACT Math : How to find the endpoints of a line segment

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Midpoint Formula

Example Question #1 : Midpoint Formula

Example Question #2 : Midpoint Formula

Example Question #3 : Midpoint Formula

Example Question #3 : Midpoint Formula

Example Question #6 : Midpoint Formula

All ACT Math Resources

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