Question

The area of a right-angled triangle is 600 cm². If the base of the triangle exceeds the altitude by 10 cm, find the dimensions of the triangle.

Answer 1

Area of a triangle = `("Height" xx "Base")/2`

Here, Height and base are x and (x + 10) and the area is 600 

Hence, `x xx (x + 10) xx 1/2 = 600`

⇒ x² + 10x = 1200 

⇒ x² + 10 x – 1200 = 0 

⇒ x² + 40x – 30x – 1200 = 0 

⇒ x(x + 40} – 30(x + 40) = 0 

⇒ (x + 40)(x – 30) = 0, hence x = 30.

Base = 30 + 10 = 40 cms.

Hence h²= 40² + 30² = 2500  

Answer 2

Let the altitude of the triangle be x cm 

Therefore, the base of the triangle will be (x + 10) cm

Area of triangle = `1/2x (x + 10) = 600`

⇒ (x + 10) = 1200

⇒ x² + 10x – 1200 = 0

⇒ x² + (40 – 30)x – 1200 = 0

⇒ x² + 40x – 30x – 1200 = 0

⇒ x(x + 40) – 30(x + 40) = 0

⇒ (x + 40) (x – 30) = 0

⇒ x = –40 or x = 30

⇒ x = 30   ...[∵ Altitude cannot be negative]

Thus, the altitude and base of the triangle are 30 cm and (30 + 10 = 40) cm, respectively.

(Hypotenuse)² = (Altitude)² + (Base)²

⇒ (Hypotenuse)² = (30)² (40)²

⇒ (Hypotenuse)² = 900 + 1600 = 2500

⇒ (Hypotenuse)² = (50)²

⇒ (Hypotenuse) = 50

Thus, the dimensions of the triangle are: 

Hypotenuse = 50 cm

Altitude = 30 cm

Base = 40 cm