Question

The length of the hypotenuse of a right-angled triangle exceeds the length of the base by 2 cm and exceeds twice the length of the altitude by 1 cm. Find the length of each side of the triangle. 

The length of the hypotenuse of a right-angled triangle exceeds the length of the base by 2 cm and exceeds twice the length of the altitude by 1 cm. Find the length of all sides. 

Answer 1

Let the base and altitude of the right-angled triangle be x and y cm, respectively.

Therefore, the hypotenuse will be (x + 2) cm. 

∴ (x + 2)² = y² + x²   ...(i) 

Again, the hypotenuse exceeds twice the length of the altitude by 1 cm. 

∴ h = (2y + 1) 

⇒ x + 2 = 2y + 1 

⇒ x = 2y – 1 

Putting the value of x in (i), we get: 

(2y – 1 + 2)² = y² + (2y – 1)² 

⇒ (2y + 1)² = y² + 4y² – 4y + 1 

⇒ 4y² + 4y + 1 = 5y² – 4y + 1 

⇒ –y² + 8y = 0

⇒ y² – 8y = 0 

⇒ y(y – 8) = 0 

⇒ y = 8 cm 

∴ x = 16 – 1

= 15 cm 

∴ h = 16 + 1

= 17 cm 

Thus, the base, altitude and hypotenuse of the triangle are 15 cm, 8 cm and 17 cm, respectively.