Question

Is M the midpoint of AB? (A is at (-4,5), M is at (4,9) and B is at (12,13)).

Answer 1

ANSWER

Yes, M is the midpoint of AB

EXPLANATION

To know if M is the midpoint of AB, we have to see if the distance from A to M is the same distance from M to B.

The distance between two points (x1, y1) and (x2, y2) is:

$d=\sqrt[]{(x_1-x_2)^2+(y_1-y_2)^2}$

The distance AM is:

$\begin{gathered} d_{AM}=\sqrt[]{(-4-4)^2+(5-9)^2} \\ d_{AM}=\sqrt[]{8^2+4^2} \\ d_{AM}=\sqrt[]{64+16} \\ d_{AM}=\sqrt[]{80} \\ d_{AM}=4\sqrt[]{5} \end{gathered}$

The distance MB is:

$\begin{gathered} d_{MB}=\sqrt[]{(4-12)^2+(9-13)^2} \\ d_{MB}=\sqrt[]{8^2+4^2} \\ d_{MB}=\sqrt[]{64+16} \\ d_{MB}=\sqrt[]{80} \\ d_{MB}=4\sqrt[]{5} \end{gathered}$

Then

$d_{AM}=d_{MB}$

Therefore M is the midpoint of AB