The perimeter of right angled triangle …

Let shortest side be x units and other side be y units 

 Hypotenuse = z units
As per given condition
The perimeter of right-angled triangle is five times the length of its shortest side.
So,  x + y + z = 5x
{tex}\Rightarrow{/tex}y + z= 4x
{tex}\Rightarrow{/tex} z  = 4x - y ...(i)
The numerical value of the area of the triangle is 15 times the numerical value of the length of the shortest side.
So, area of the rectangle is = 15x
{tex}\Rightarrow{/tex}{tex}\frac{1}{2}{/tex}x{tex}\cdot{/tex}y = 15x 
{tex}\Rightarrow{/tex} y = 30 ....(ii)
Using Pythagoras Theorem, we get
z² = x² + y²
{tex}\Rightarrow{/tex}(4x - y)² = x² + y² [from (i)]
{tex}\Rightarrow{/tex}(4x - 30)² = x² + (30)²  [Using (ii)]
{tex}\Rightarrow{/tex}16x² - 240x + 900 = x² + 900
{tex}\Rightarrow{/tex}15x² - 240x = 0
{tex}\Rightarrow{/tex}15x(x - 16) = 0
{tex}\Rightarrow{/tex}x = 0(rejecting) or x = 16
{tex}\therefore{/tex}x = 16
{tex}\therefore{/tex}length of the shortest side = 16 units
length of other side = 30 units
length of hypotenuse z = 4 {tex}\times{/tex} 16 - 30 = 34 units