Determine whether the triangle having sides $(a - 1)\ cm$, $2\sqrt{a}\ cm$ and $(a + 1)\ cm$ is a right angled triangle.

Given:

The lengths of the sides of a triangle are $(a - 1)\ cm$, $2\sqrt{a}\ cm$ and $(a + 1)\ cm$.
To do:
We have to determine whether the given triangle is a right-angled triangle. 

Solution:

Let $AB=(a - 1)\ cm$, $BC=2\sqrt{a}\ cm$ and $CA=(a + 1)\ cm$.

We know that,

In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.

Therefore,

$AB^=(a-1)^2\ cm^2$

$=a^2-2(a)(1)+(1)^2\ cm^2$

$=a^2-2a+1\ cm^2$

$BC^2=(2\sqrt{a})^2\ cm^2$

$=4a\ cm^2$

$CA^2=(a+1)^2\ cm^2$

$=a^2+2(a)(1)+(1)^2\ cm^2$

$=a^2+2a+1\ cm^2$

$AB^2+BC^2=(a^2-2a+1)+(4a)\ cm^2$

$=a^2+2a+1\ cm^2$

This implies,

$AB^2+BC^2=CA^2$

Therefore, the given triangle is a right-angled triangle.

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Updated on: 10-Oct-2022

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