Question

The mid-point of the sides AB and AC of $△ABC$ are (3, 5) and (-3, -3) respectively. What is the length of the side BC?

• A. $10$
• B. $20$
• C. $15$
• D. $30$

Answer 1

The correct option is B $20$

$\because M$ & $N$ are the mid-point of $AB$ and $AC$.

$\therefore MN\parallel BC\text{(mid-point theorem)}$

$\therefore$ In $△AMN$ & $△ABC$

$\angle A=\angle A$ (common)

$\angle AMN=\angle ABC$ (corresponding angles)

$\angle ANM=\angle ACB$ (corresponding angles)

$\therefore △AMN\sim △ABC$ $\left(AAA\right)$

$\therefore \frac{AM}{AB}=\frac{MN}{BC}=\frac{AN}{NC}=\frac{1}{2}$

$\therefore \frac{MN}{BC}=\frac{1}{2}$

$BC=2MN$

$\therefore MN=\sqrt{{(-3-3)}^2+{(-3-5)}^2}=\sqrt{36+64}=10$

$\therefore BC=2\times 10=20$