Question

Perimeter of right angle is 144cm and its hypotenuse is 65 cm find the other two sides and calculate the area and verify the result using herons formula

Answer 1

the two sides are x and y.

Perimeter of right angled triangle is 144 cm.

Step 2:

So we get

And   x + y + 65 = 144

        x + y   =  144 - 65 = 79

Step 3:

Squaring both sides, we get

4225 + 2 x y = 6241

2 x y  = 6241 – 4225

2 x y      = 2016

x y  =   2016 / 2

x y =   1008

y     =    1008 / x

Step 4:

Sub:      

y     =    1008 / x    in x + y = 79, we get  

x + 1008 / x   =  79

  +   1008   =  79 x  

  -  79 x  +1008  =  0  

   - 63 x - 16 x  + 1008  = 0  

x ( x - 63 )  -  16 ( x - 63 )  = 0

 ( x - 63 ) ( x - 16 ) =  0

So, x = 63 cm  or  x  =  16 cm

Step 5:

The length of other sides are 63 cm, 16 cm.

Area of right angled triangle is  

1 / 2 x 63 x 16 = 504cm²

Step 6:

Verify the result using Heron's Formula

S = ( 16 + 63 + 65 ) / 2  = 72

A = ( s x ( s-a ) x (  s - b ) x ( s - c ) )1/2

      = ( 72 x (72 - 16 ) x ( 72 - 63 ) x ( 72 - 65 )  )1/2

     =   ( 72 x 56 x 9 x 7 ) 1/2

    =   ( 9 x 8 x 8 x 7 x 9 x 7 )1/2                

A    =   9 x 8 x 7 = 504.cm²

Answer 2

${\textbf{Answer}}$

The perimeter of right angled triangle is 144 cm.

 hypotenuous is 65 cm .

 let the two sides are x and y .

 x 2 + y2 = 652        

Perimeter of right angled triangle is 144 cm.

And   x + y + 65 = 144

         x + y   =  144 - 65 = 79

squaring both sides, we get

            ( x + y )2 = 792

=>      x2 + y2 + 2 x y = 6241

           4225 + 2 x y = 6241

           2 x y  = 6241 - 4225

                     = 2016

              x y  =   2016 / 2

                     =   1008

              y     =    1008 /  x

  y     =    1008 / x    in  x  + y  = 79  , we get

                 x  +  1008 / x   =  79

                 x2  +   1008   =  79 x

                 x2  -  79 x  +1008  =  0

                 x2   - 63 x - 16 x  + 1008  = 0

                x ( x - 63 )  -  16 ( x - 63 )  =0

                 ( x - 63 ) ( x - 16 ) =  0

                 so, x = 63 cm  or  x  =  16 cm

Here the length of other sides are 63 cm , 16 cm.

Area of  right  angled triangle is                  1 / 2  x  63 x 16 = 504 cm2

 Verifying the result using Heron's Formula

            S = ( 16 + 63 + 65 ) / 2  = 72

   A = ( s x ( s-a ) x (  s - b ) x ( s - c ) )1/2

       = ( 72 x (72 - 16 ) x ( 72 - 63 ) x ( 72 - 65 )  )1/2

      =   ( 72 x 56 x 9 x 7 ) 1/2

     =   ( 9 x 8 x 8 x 7 x 9 x 7 )1/2                

     =   9 x 8 x 7 = 504 cm2 

9 x 8 x 7 = 504 cm2 Result is verified.