Question

Prove that a triangle with the sides a - 1 cm to root under a ​

Answer 1

Question is Incomplete, Complete question is given below.

Prove that a triangle with the sides (a − 1) cm, 2√a cm and (a + 1) cm is a right angled triangle.

Answer:

∆ABC is right angled triangle with right angle at B.

Explanation:

Given : Triangle having sides (a - 1) cm, 2√a and (a + 1) cm.

We need to prove that triangle is the right angled triangle.

Let the triangle be denoted by Δ ABC with side as;

AB = (a - 1) cm

BC = (2√ a) cm

CA = (a + 1) cm

Hence,

`{AB}^2 = (a -1)^2`

Now We know that

`(a- b)^2 = a^2+b^2 - 2ab`

So;

`{AB}^2= a^2 + 1^2 -2* a *1`

`{AB}^2 = a^2 + 1 -2a`

Now;

`{BC}^2 = (2√(a))^2= 4a`

Also;

`{CA}^2 = (a + 1)^2`

Now We know that

`(a+ b)^2 = a^2+b^2 + 2ab`

`{CA}^2= a^2 + 1^2 +2* a *1`

`{CA}^2 = a^2 + 1 +2a`

`{CA}^2 = AB^2 + BC^2`

[By Pythagoras theorem]

`a^2 + 1 +2a = a^2 + 1 - 2a + 4a\\\\a^2 + 1 +2a= a^2 + 1 +2a`

Hence,
`{CA}^2 = AB^2 + BC^2`

Now In right angled triangle the sum of square of two sides of triangle is equal to square of the third side.

This proves that ∆ABC is right angled triangle with right angle at B.