Question

In the given figure $ABC$ is a right triangle and right angled at $B$ such $\angle BCA=2\angle BAC$.
Show that hypotenuse $AC=2BC$.
(Hint: Produce $CB$ to a point $D$ that $BC=BD$)

Answer 1

$ABC$ is right angled triangle at $B$

$\angle BCA=2\angle BAC$

Show that hypotenuse $AC=2BC$

We have produced $CB$ to $D$ such that $BC=BD$

$△ABC$ & $△ABD$ are congruent triangle because

$AB=AB$

$\angle ABC=\angle ABD={90}^o$

$BC=BD$ (given)

by $SAS$ property

$\angle BAC=\angle BAD$

$\angle BAC+\angle BAD=2\angle BAC=\angle CAD$

$\angle BCA=2\angle BAC$ (given)

$\angle CAD=\angle BCA\to \angle CAD=\angle BDA$

In $△ACD$, sides opposite to equal angles are equal

$CD=AC$

$BC+BD=AC$

$2BC=AC(\because BC=BD)$

Hence Proved.