This question has been taken from Book:- ML aggarwal, Avichal publication, class10th, quadratic equation in one variable, chapter 5, exercise 5.5

This is an important ques

If the sum of two smaller sides of a right-angled triangle is 17cm and the perimeter is 30cm, then find the area of the triangle.

Question no.19 , ML Aggarwal, chapter 5, exercise 5.5, quadratic equation in one variable, ICSE board,

Given,

The perimeter of the triangle = 30 cm

Let’s assume the length of one of the two small sides as x cm

Then, the other side will be (17 – x) cm

Now, length of hypotenuse = perimeter – sum of other two sides

= (30 – 17) cm

= 13 cm

According to the problem, by Pythagoras theorem we have

x

^{2}+ (17 – x)^{2}= 13^{2}x

^{2}+ 289 + x^{2}– 34x = 1692x

^{2}– 34x + 289 – 169 = 02x

^{2}– 34x + 120 = 0x

^{2}– 17x + 60 = 0 [Dividing by 2]By factorization method, we have

x

^{2}– 12x – 5x + 60 = 0x(x – 12) – 5(x – 12) = 0

(x – 5) (x – 12) = 0

So,

(x – 5) = 0 or (x – 12) = 0

x = 5 or x = 12

When, x = 5

First side = 5 cm and second side = (17 – 5) = 12 cm

And when x = 12

First side = 12 cm and second side = (17 – 12) = 5 cm

Thus,

Area of the triangle = ½ (5 × 12)

= 60/2

= 30 cm

^{2}