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Algebra 1 : Functions and Lines

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Example Questions

Example Question #4 : Midpoint Formula

Line segment XY has a midpoint of (6,3) . If X is (2,8) what is Y?

Possible Answers:

(7,4)

(10,-2)

(10,2)

(2,6)

(4,5.5)

Correct answer:

(10,-2)

Explanation:

For this kind of problem, it's important to keep in mind how midpoint is solved for:

$midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) = (x_m,y_m)$ where is the midpoint coordinate.

Because we've been given the midpoint already, we'll have to solve this problem backwards. Since we've been given one of the endpoints (X) and we just need to solve for the other end point (Y), we may arbitrarily assign (2,8) as (x_1,y_1) . If we substitute in the two coordinates - an endpoint and the midpoint - we should be able to solve for the unknown endpoint.

$(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) = (x_m,y_m)$

$(\frac{2+x_2}{2},\frac{8+y_2}{2}) = (6,3)$

It may be visually easier to break the arithmetic into separate operations.

$\frac{2+x_2}{2}=6$ and $\frac{8+y_2}{2}=3$

By separating the x and y components, we can easily solve for the missing endpoint now.

$2[\frac{2+x_2}{2}=6]$

2+x_2=12

$x_2={\color{Blue} 10}$

Doing similar arithmetic, y_2 will be solved to be .

Therefore, endpoint Y is (10,-2)

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Example Question #2 : Midpoint Formula

Line segment EF has a midpoint of (7,6) . If endpoint F is at (2,3) , what's the coordinate for endpoint E?

Possible Answers:

(12,7)

(12,9)

(10,12)

(-11,3)

(7,4)

Correct answer:

(12,9)

Explanation:

For this kind of problem, it's important to keep in mind how midpoint is solved for:

$midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) = (x_m,y_m)$ where is the midpoint coordinate.

Because we've been given the midpoint already, we'll have to solve this problem backwards. Since we've been given one of the endpoints (F) and we just need to solve for the other end point (E), we may arbitrarily assign (2,3) as (x_1,y_1) . If we substitute in the two coordinates - an endpoint and the midpoint - we should be able to solve for the unknown endpoint.

$(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) = (x_m,y_m)$

$(\frac{2+x_2}{2},\frac{3+y_2}{2}) = (7,6)$

It may be visually easier to break the arithmetic into separate operations.

$\frac{2+x_2}{2}=7$ and $\frac{3+y_2}{2}=6$

By separating the x and y components, we can easily solve for the missing endpoint now.

$2[\frac{2+x_2}{2}=7]$

2+x_2=14

$x_2={\color{Blue} 12}$

Doing similar arithmetic, y_2 will be solved to be .

Therefore, endpoint E is (12,9) .

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Example Question #3 : Midpoint Formula

Line segment DF has a midpoint of (4,8) . If endpoint D is at (22,7) , where is endpoint F?

Possible Answers:

(30,9)

(-14,9)

(5,-9)

(11,-18)

(22,13)

Correct answer:

(-14,9)

Explanation:

For this kind of problem, it's important to keep in mind how midpoint is solved for:

$midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) = (x_m,y_m)$ where is the midpoint coordinate.

Because we've been given the midpoint already, we'll have to solve this problem backwards. Since we've been given one of the endpoints (D) and we just need to solve for the other end point (F), we may arbitrarily assign (22,7) as (x_1,y_1) . If we substitute in the two coordinates - an endpoint and the midpoint - we should be able to solve for the unknown endpoint.

$(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) = (x_m,y_m)$

$(\frac{22+x_2}{2},\frac{7+y_2}{2}) = (4,8)$

It may be visually easier to break the arithmetic into separate operations.

$\frac{22+x_2}{2}=4$ and $\frac{7+y_2}{2}=8$

By separating the x and y components, we can easily solve for the missing endpoint now.

$2[\frac{22+x_2}{2}=4]$

22+x_2=8

$x_2={\color{Blue} -14}$

Doing similar arithmetic, y_2 will be solved to be .

Therefore, endpoint Y is (-14,9) .

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Example Question #7 : Midpoint Formula

The midpoint of a line segment is represented by the point $\left(\frac{15}{2},1\right)$ . If the coordinates for one of its endpoints are (2,10) and the y-coordinate of the other endpoint is 5, find the value of the x-coordinate. To clarify, our endpoints are (2,10) and (?,5)

Possible Answers:

x=-1

x=3

x=0

x=2

x=1

Correct answer:

x=0

Explanation:

We know that the midpoint of our line segment is $\left(\frac{15}{2},1\right)$ . To find the x-coordinate of this segment, we work backwards, starting with our midpoint formula. In this case, we only need to use the midpoint formula to solve for the x-coordinate, which looks like:

$1=\frac{2-x}{2}$

Next, multiply both sides of the equation by 2, which gives us:

2=2-x , which means our missing x-coordinate is 0. So, the endpoints of our line segment are $(0,5)\&(2,10)$ .

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

If the midpoint of a line segment is (3, 4) and one endpoint is (-1, 2), find the other endpoint.

Possible Answers:

(4, 2)

(3, 8)

(4, 6)

(2, 6)

(7, 6)

Correct answer:

(7, 6)

Explanation:

To solve, we will using the midpoint formula and substitute what we know. The midpoint formula is:

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

where (x_1, y_1) and (x_2, y_2) are the endpoints.

Now, here is what we know:

$\text{midpoint} = (3, 4)$

(x_1, y_1) = (-1, 2)

Here is what we are solving for

$(x_2, y_2) = \ ???$

So, we will substitute. We get

$(3, 4) = (\frac{-1 + x_2}{2}, \frac{2 + y_2}{2})$

We can divide this into parts. We know

$3 = \frac{-1 + x_2}{2}$

and

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

So, we can solve for x_2 and y_2 to find the other endpoint.

$3 = \frac{-1 + x_2}{2}$

$3 \cdot 2 = \frac{-1 + x_2}{2} \cdot 2$

6 = -1 + x_2

6 + 1 = -1 + x_2 + 1

7 = x_2

x_2 = 7

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

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

8 = 2 + y_2

8 - 2 = 2 + y_2 - 2

6 = y_2

y_2 = 6

This give us the point (7, 6) . Therefore, the other endpoint is (7, 6) .

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Example Question #1 : Midpoint Formula

A line segment has the midpoint (-2,3) . One end point of the line segment is located at (5,6) . What is the other end point?

Possible Answers:

(-9,0)

(-1,12)

(1,12)

(9,0)

Correct answer:

(-9,0)

Explanation:

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

$(-2,3)=(\frac{5+x_2}{2},\frac{6+y_2}{2})$

$2(-2,3)=2(\frac{5+x_2}{2},\frac{6+y_2}{2})$

(-4,6)=(5+x_2,6+y_2)

(x_2,y_2)=(-4-5,6-6)

(x_2,y_2)=(-9,0)

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Example Question #2 : Midpoint Formula

Find the midpoint on the line segment from (2, 3) to (4, 1).

Possible Answers:

(–2, 2)

(3, 2)

(6, 4)

(2, 2)

(2, –2)

Correct answer:

(3, 2)

Explanation:

By using the midpoint formula, we can find the x and y coodinantes fo the midpoint.

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

$\frac{x_1+x_2}{2}=\frac{2+4}{2}=\frac{6}{2}=3$

$\frac{y_1+y_2}{2}=\frac{3+1}{2}=\frac{4}{2}=2$

Our coordinates are (3, 2).

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

Point X (2, 9) and Point Y (8, 3) are endpoints on a line segment. What is the Midpoint M of that line segment?

Possible Answers:

$\left ( 6,6 \right )$

$\left ( 6,7 \right )$

$\left ( -3,6 \right )$

$\left ( 2,8 \right )$

$\left ( 5,6 \right )$

Correct answer:

$\left ( 5,6 \right )$

Explanation:

To find the midpoint of a line segment, you add together the components and divide by two ( $\frac{2+8}{2}$ = 5) , do the same for ( $\frac{3+9}{2}$ =6). The answer is (5, 6).

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

What is the midpoint of the points (3,12) and (9,15)?

Possible Answers:

(7,13)

(1.5,7.5)

(8,12)

(6,13.5)

Correct answer:

(6,13.5)

Explanation:

To find the midpoint we must know the midpoint formula which is

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

We then take the -coordinate from the first point and plug it into the formula as $x_{1}$ .

We take the -coordinate from the second point and plug it into the formula as $x_{2}$ .

We then do the same for $y_{1}$ and $y_{2}$ .

With all of the points plugged in our equation will look like this.

$(\frac{3+9}{2},\frac{12+15}{2})$

We then perform the necessary addition and division to get the answer of

$(\frac{12}{2},\frac{27}{2})=(6,13.5)$

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

Find the midpoint of the line segment that connects the two points below.

Point 1: (0,-3)

Point 2: (-2, 4)

Possible Answers:

$(-1, \frac{7}{2})$

$(-1, \frac{1}{2})$

$(1, -\frac{1}{2})$

$(-1,-\frac{1}{2})$

$(1, -\frac{7}{2})$

Correct answer:

$(-1, \frac{1}{2})$

Explanation:

The average of the the -coordinates and the average of the y-coordinates of the given points will give you the mid-point of the line that connects the points.

$\textup{midpoint}=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$ , where (x_1,y_1) is (0,-3) and (x_2,y_2) is (-2, 4) .

$(\frac{(0)+(-2)}{2}, \frac{(-3)+(4)}{2})$

$(\frac{-2}{2},\frac{1}{2})$

$(-1, \frac{1}{2})$

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Algebra 1 : Functions and Lines

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #4 : Midpoint Formula

Example Question #2 : Midpoint Formula

Example Question #3 : Midpoint Formula

Example Question #7 : Midpoint Formula

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

Example Question #1 : Midpoint Formula

Example Question #2 : Midpoint Formula

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

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

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

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