Question

A line segment has an endpoint at (3,2). if the midpoint of the line segment is (6,-2), what are the coordinates of the point at the other end of the line segment?

Answer 1

We are given that the end-point of a line segment is (3, 2) and the middle point is (6, -2).

The formula for the middle point of a line segment is given by:

`(h,k)=((x_1+x_2)/(2),(y_1+y_2)/(2))`

Where "h" and "k" are the middle points of the segment.

Now, we can set each of the coordinates equal. For the x-coordinates we have:

`h=(x_1+x_2)/(2)`

Where:

`x_1,x_2=\text{ coordinates of the end-points}`

Now, we solve for the second end-point.

First, we multiply both sides by 2:

`2h=x_1+x_2`

Now, we subtract x1 from both sides:

`2h-x_1=x_2`

Now, we substitute both sides:

`\begin{gathered} 2(6)-3=x_2 \\ 12-3=x_2 \\ 9=x_2 \end{gathered}`

Now, we solve for the y-coordinate:

`k=(y_2+y_1)/(2)`

Now, we multiply both sides by 2 and subtract the first coordinate to both sides:

`2k-y_1=y_2`

Now, we plug in the values:

`\begin{gathered} 2(-2)-2=y_2 \\ -4-2=y_2 \\ -6=y_2 \end{gathered}`

Therefore, the coordinates of the other end-point is:

`(x_2,y_2)=(9,-6)`