How to find the endpoints of a line segment - PSAT Math

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).

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).

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.

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:

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:

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:

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:

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).

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).

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).

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.

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:

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:

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:

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:

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).

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).

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).

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.