Question

Find the coordinates of A if M(-1, 2) is the midpoint of AB and B has coordinates of (3,-5).

Answer 1

Hello there. To solve this question, we have to remember how to determine the coordinates of a point given the midpoint of a segment.

Given that M is the midpoint of the segment AB

$M=(-1,\,2)$

and that the point B has coordinates

$B=(3,\,-5)$

First, remember the formula for the distance between two points (x0, y0) and (x1, y1):

$d((x_0,\,y_0),\,(x_1,\,y_1))=\sqrt{(x_0-x_1)^2+(y_0-y_1)^2}$

So that we know that the midpoint of a segment has the same distance from its ends.

In this case, we determine first the distance between the points M and B:

$\begin{gathered} d(M,\,B)=\sqrt{(-1-3)^2+(2-(-5))^2}=\sqrt{(-4)^2+7^2} \\ \\ \Rightarrow d(M,\,B)=\sqrt{16+49}=\sqrt{65} \\ \end{gathered}$

Next step, remember the formula for the midpoint of a segment

If A and B are the endpoints of the segment AB and has coordinates

$A=(x_A,\,y_A)\text{ and }B=(x_B,\,y_B)$

Its midpoint is given by

$M=\left(\dfrac{x_A+x_B}{2},\,\dfrac{y_A+y_B}{2}\right)$

Such that we find

$M=(-1,\,2)=\left(\dfrac{x_A+3}{2},\,\dfrac{y_A-5}{2}\right)$

Hence we find that

$\begin{gathered} \dfrac{x_A+3}{2}=-1\Rightarrow x_A=-5 \\ \\ \dfrac{y_A-5}{2}=2\Rightarrow y_A=9 \end{gathered}$

So the coordinates of the point A are

$A=(-5,\,9)$