Question

The midpoint of a segment is (−2,−3) and one endpoint is (3,0) . Find the coordinates of the other endpoint.

Answer 1

To work out the mid point of two points you, add the x coordinates and divide by 2, and you take the y coordinates and divide by two:

So:

$midpoint = \frac{sum.of.x-coords}{2}, \frac{sum.of.y-coords}{2}$

------------------------------------

So the x-coords of the midpoint is:

$\frac{sum.of.x-coords}{2}$

and

y -coords of midpoint is:

$\frac{sum.of.y-coords}{2}$

------------------------------------

However, in this question we are trying to work out one of the endpoints.

First let's say that the coordinates of the missing endpoint is:

(x , y)

_____________________________________________

That means that the x-coords of the midpoint of (x, y) and the other endpoint (3, 0) is :

$\frac{3 + x}{2}$

However, we already know the x-coord of the midpoint ( it's -2). So we can form an equation to workout x:

$\frac{3 + x}{2} = -2$                (multiply both sides by 2)

$3 + x = -4$                      (subtract 3 from both sides)

$x = -7$

This is the x-coord of the other endpoint

_______________________________________________

Let's do the same for the y coordinates:

We know y coords for the midpoint of (x, y) and (3, 0) is:

$\frac{0 + y}{2}$

But we also know the ycoord is -3. So we can form an equation and solve for y:

$\frac{0+y}{2} = -3$

$\frac{0 + y}{2} = -3$   (multiply both sides by 2)

$0 + y = -6$                (simplify)

$y = -6$

This is the y-coord of the other endpoint

___________________________________

Now we just put these coords together to get the coordinate of the other endpoint:

Endpoint is at:

(x, y)                           (substitute in values that we worked out)

=  (-7, -6)

_________________________________________________

Answer:

D. (-7, -6)

________________________________________________

Note:

If there is anything you don't quite understand or was unclear

- please don't hesitate to ask below in the comments.

Answer 2

The midpoint can be defined using formula,

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

So by knowing $x_m, x_1$ and $y_m, y_1$ we can calculate $x_2, y_2$

First we must derive two equations,

$x_m=\dfrac{x_1+x_2}{2}\Longrightarrow x_2=2x_m-x_1$

and

$y_m=\dfrac{y_1+y_2}{2}\Longrightarrow y_2=2y_m-y_1$

Then just put in the data,

$x_2=2\cdot(-2)-3=-7$

$y_2=2\cdot(-3)-0=-6$

So the other endpoint has coordinates $(x,y)\Longrightarrow(-7, -6)$ therefore the answer is D.

Hope this helps.

r3t40