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The perimeter of right angled triangle …

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The perimeter of right angled triangle is five times the length of its shortest side the numerical value of the area of the triangle is 15times the numerical value of the length of the shortest side find the lengths of the three sides of the triangle
  • 1 answers

Sia ? 6 years, 5 months ago

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) [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

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