Question

The sides of a right angled triangle are in arithmetic progression. If the triangle has area $24,$ then what is the length of its smallest side?

Answer 1

Let the sides be,

$\Rightarrow$$a-d,a,a+d$

Given,

using Pythagorus theorem,

$\Rightarrow$$(a-d{)}^2+a^2=(a+d{)}^2$

$\Rightarrow$$a^2+d^2-2ad+a^2=a^2+d^2+2ad$

$\Rightarrow$$a^2=4ad$

$\Rightarrow a=4d$

Therefore the sides are:

$\Rightarrow$$3d,4d,5d$

$\Rightarrow$$Area=\frac{1}{2}\times 3d\times 4d$

$\Rightarrow$$24=6d^2$

$\therefore d^2=4\Rightarrow d=2$

$\Rightarrow$we get, $6,8,10$

$\Rightarrow$The smallest sides is $6$