Question

If the angles of a right angled triangle are in $A.P$., then ratio of its sides is

• A. $1:\sqrt{3}:2$
• B. $1:1:\sqrt{2}$
• C. $2:3:\sqrt{13}$
• D. $1:2:3$

Answer 1

Answer: (A)

$1:\sqrt{3}:2$

Let the angles of the right triangle be $a-d,a$ and $a+d$ where $d$ is the common difference.

Then, $a+(a+d)+(a-d)=180$.

That gives, $a=60$.

One of the angle is $90$ degrees (right angled triangle), i.e., $a+d=90$ or $d=30$. Therefore, the angles are ${30}^{\circ }$, ${60}^{\circ }$ and ${90}^{\circ }$.

Using the sine rule,

$\frac{a}{\sin A}=$ $\frac{b}{\sin B}=$ $\frac{c}{\sin C}=k$

$a=k\sin {30}^{\circ }=\frac{k}{2}$

$b=k\sin {60}^{\circ }=\frac{\sqrt{3}k}{2}$

$c=k\sin {90}^{\circ }=k$

Therefore, $a:b:c=$ $1:\sqrt{3}:2$. Option (A) is correct.