Question

In the given figure $ABC$ is a right triangle and right angled at $B$ such that $\angle BCA=2\angle BAC$. The value of $AC-2BC$ is:

Answer 1

$△ABC$ is a right angled triangle at $\angle B$

$\angle B={90}^o$

Let $\angle BAC=x$

then $\angle BCA=2x$

Thus,

$\angle B+\angle BAC+\angle BCA={180}^o$

$90+x+2x=180$

$90+3x=180$

$3x=90$

$x=30$

then,

$\angle BAC={30}^o$ and $\angle BCA=2\left(30\right)={60}^o$

Therefore the measure of angles of triangle ABC are 30,60 and 90

Here side AC is a hypotenuse

therefore

Side opposite to ${30}^o$ is half the hypotenuse

here the side opposite to ${30}^o$ is side BC

therefore

$BC=\frac{1}{2}\times AC$

thus

$AC=2BC$