Question

The sides of a right angled $△$ are in A.P. If the $△$ has area $24$, then what is the length of its side?

Answer 1

Let $x$ be the base, $x+d$ be the hypotenuse and $x-d$ be the other side
Then,

$\frac{1}{2}(x-d)\left(x\right)=24$

By Pythagoras theorem,

$(x-d{)}^2+(x{)}^2=(x+d{)}^2$

$x^2+x^2+d^2-2xd=x^2+d^2+2xd$

$x^2=4xd$

$x=4d$

Therefore, the sides of triangle become $3d,4d,5d$

Now using the above equation

$\frac{1}{2}\left(3d\right)\left(4d\right)=24$

$d^2=4$

$d=2$

The smallest side will be $3d=3\times 2=6$