Question

the sides of an right angled triangle containing the right angle are 5x cm and (3x-1)cm
calculate the length of the hypotenuse of the triangle if it's area is 60cm^2

Answer 1

Given that sides of a right-angled triangle are 5x and (3x - 1)cm.

Given that Area of the triangle = 60cm^2.

We know that Area of the triangle = 1/2 * b * h

                              60 = 1/2 * 5x * (3x - 1)

                             5x(3x - 1) = 60 * 2

                             5x(3x - 1) = 120

                             x(3x - 1) = 120/5

                              3x^2 - x = 24

                              3x^2 - x - 24 = 0

                              3x^2 + 8x - 9x - 24 = 0

                              x(3x + 8) - 3(3x + 8)

                              (x - 3)(3x + 8)

                              x = 3 (or) x = -3/8.


x value should not be -ve.Therefore the value of x = 3.

Therefore the sides of a right-angled triangle = 

5x = 5 * 3 = 15cm

(3x - 1) = (3 * 3 - 1)

           = 9 - 1

           = 8cm

 
By Pythagoras theorem, we know that 

h^2 = 15^2 + 8^2

       = 225 + 64

       = 289

h = $\sqrt{289}$

   = 17.


Therefore the hypotenuse = 17cm.

Therefore the sides of the triangle are 8cm,15cm, and 17cm.

Hope this helps!

Answer 2

Height=5x cm
Base=(3x-1) cm
Area of triangle=1/2(b*h)
60=1/2[(3x-1)*5x]
60*2=(3x-1)*5x
​120=15x^2-5x
15x^2-5x-120=0
Dividing both sides by 5,
3x^2-x-24=0
3x^2-9x+8x-24=0
3x(x-3)+8(x-3)=0
(x-3)=0,(3x+8)=0
x=3,x=-8/3
Since length cannot be -ve,so x=3.
Height=5x=15 cm , Base=3x-1=8 cm
By pythagoras theorem,
                          =(8)^2+(15)^2
                          =64+225
                          =289
hypotenuse=17 cm