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
x2 + (17 – x)2 = 132
x2 + 289 + x2 – 34x = 169
2x2 – 34x + 289 – 169 = 0
2x2 – 34x + 120 = 0
x2 – 17x + 60 = 0 [Dividing by 2]
By factorization method, we have
x2 – 12x – 5x + 60 = 0
x(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 cm2