# PSAT Math : How to find the endpoints of a line segment

## Example Questions

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

The midpoint of line segment AB is (2, -5).  If the coordinates of point A are (4, 4), what are the coordinates of B?

(0, -13)

(0, -14)

(6, 14)

(6, 13)

(3, -0.5)

(0, -14)

Explanation:

The fastest way to find the missing endpoint is to determine the distance from the known endpoint to the midpoint and then performing the same transformation on the midpoint.  In this case, the x-coordinate moves from 4 to 2, or down by 2, so the new x-coordinate must be 2-2 = 0.  The y-coordinate moves from 4 to -5, or down by 9, so the new y-coordinate must be -5-9 = -14.

An alternate solution would be to substitute (4,4) for (x1,y1) and (2,-5) for (x,y) into the midpoint formula:

x=(x1+x2)/2

y=(y1+y2)/2

Solving each equation for (x2,y2) yields the solution (0,-14).

### Example Question #1 : Midpoint Formula

Point A is (5, 7).  Point B is (x, y).  The midpoint of AB is (17, –4). What is the value of B?

(8.5, –2)

(29, –15)

(12, –11)

(22, –9)

(29, –15)

Explanation:

Point A is (5, 7). Point B is (x, y). The midpoint of AB is (17, –4). What is the value of B?

We need to use our generalized midpoint formula:

MP = ( (5 + x)/2, (7 + y)/2 )

Solve each separately:

(5 + x)/2 = 17 → 5 + x = 34 → x = 29

(7 + y)/2 = –4 → 7 + y = –8 → y = –15

Therefore, B is (29, –15).

### Example Question #1 : Midpoint Formula

Line segment AB has an endpoint, A, located at , and a midpoint at . What are the coordinates for point B of segment AB?

The second endpoint cannot exist

Explanation:

With an endpoint A located at (10,-1), and a midpoint at (10,0), we want to add the length from A to the midpoint onto the other side of the segment to find point B. The total length of the segment must be twice the distance from A to the midpoint.

A is located exactly one unit below the midpoint along the y-axis, for a total displacement of (0,1). To find point B, we add (10+0, 0+1), and get the coordinates for B: (10,1).

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

One endpoint of a line segment on the coordinate plane is . The segment has length 10; the other endpoint has -coordinate 10 and is in Quadrant I. Give the -coordinate of the other endpoint. (Nearest tenth if applicable).

Explanation:

Let  be the -coordinate of the other endpoint.

The segment has endpoints  and .

Apply the distance formula

setting ,

Therefore, there are two values of  which fit the distance criterion.

One is .

However, since the endpoint is in Quadrant I, it must have a positive  coordinate, so this is eliminated as a choice.

The other is , which is correct since it is positive.

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

The midpoint of a line segment is the point . One endpoint is ; give the -coordinate of the other.

Explanation:

Let the endpoints of a line segment be

Then the midpoint of the segment will be

The -coordinate of the midpoint is , and the -coordinate of the known endpoint is , so we can solve for , the -coordinate of the unknown endpoint, in the following equation:

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

The midpoint of a line segment is the point . One endpoint is ; give the -coordinate of the other.

Explanation:

Let the endpoints of a line segment be

Then the midpoint of the segment will be

The -coordinate of the midpoint is , and the -coordinate of the known endpoint is , so we can solve for , the -coordinate of the unknown endpoint, in the following equation:

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

The midpoint of a line segment is the point . One endpoint is ; give the -coordinate of the other.

Explanation:

Let the endpoints of a line segment be

Then the midpoint of the segment will be

The -coordinate of the midpoint is , and the -coordinate of the known endpoint is

so we can solve for , the -coordinate of the unknown endpoint, in the following equation:

### Example Question #5 : How To Find The Endpoints Of A Line Segment

The midpoint of a line segment is the point . One endpoint is ; give the -coordinate of the other.