Question

Let a, b, c be the lengths of the sides of a right angled triangle, the hypotenuse having the length c, then a + b is

(a) > 2c 

(b) < √2c 

(c) > √2c 

(d) < 2c

Answer 1

(b) < √2c

Since a, b, c are the lengths of the sides of a right triangle having hypotenuse of length c, therefore a² + b² = c² 

(Pythagoras, Theoram) 

Now, a > 0, b > 0 ⇒ (a – b)² > 0 

⇒ a² + b² > 2ab 

⇒ 2(a² + b²) > a² + b² + 2ab 

⇒ 2 (a² + b²) > (a + b)² 

⇒ 2c² > (a + b)² ⇒ (a + b)² < 2c² ⇒ (a + b) < √2c.