Question

If the length of sides of a right-angled triangle are in arithmetic progression, then find their ratio?

Answer 1

Since the length of sides of the triangle are in AP, let us assume they are $a,(a+d)$ and $(a+2d)$.

By Pythagoras theorem,

$(a+2d{)}^2=(a+d{)}^2+a^2$

$a^2+4d^2+4ad=a^2+d^2+2ad+a^2$

$3d^2+2ad-a^2=0$

Assuming, $d=1$, we get

$3+2a-a^2=0$

$\Rightarrow a^2-2a-3=0$

On solving, we get $a=-1$ and $a=3$

Since, length of sides cannot be negative, $a=3$.

The other two sides are $3+1=4$ and $3+2\left(1\right)=5$.

Hence, the sides of the right-angled triangle are in the ratio $3:4:5$.