Question

Prove that the line segment joining the mid-point of the hypotenuse of a right triangle to its opposite vertex is half of the hypotenuse.

Answer 1

Given: In a right-angled $\Delta ABC$, $\angle B={90}^o$. D is the midpoint of hypotenuse AC. DB is joined.

To prove: BD = $\frac{1}{2}AC$

Construction: Draw a circle with centre D and AC as diameter.

Proof: $\because \angle ABC={90}^o$

$\therefore$ The circle drawn on AC as diameter will pass through B.

$\therefore$ BD is the radius of the circle.

But AC is the diameter of the circle and D is mid piont of AC.

$\therefore$ AD = DC = BD

$\therefore BD=\frac{1}{2}AC$