Question

The given figure shows a right triangle right angled at B If $\angle BCA=2\angle BAC,$ Then, $\frac{AC}{BC}=$

Answer 1

In $△ABC$,
$\angle BCA=2\angle ABC$
let $\angle BAC=x$
Then, $\angle BCA=2x$
Sum of angles = 180
$\angle BCA+\angle BAC+\angle ABC=180$
$2x+x+90=180$
$3x=90$
$x=30$
Thus, $\angle BCA={60}^{\circ }$ and $\angle BAC={30}^{\circ }$
Now, $\frac{BC}{AC}=Sin\angle BAC$
$\frac{BC}{AC}=Sin30$
$\frac{BC}{AC}=\frac{1}{2}$
$2BC=AC$