Verify whether the following measures represent the sides of a right angled triangle.
$\sqrt{2}, \sqrt{3}, \sqrt{5}$

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Solution

Let $△ A B C$ be a triangle where $A B = \sqrt{2}$ , $B C = \sqrt{3}$ and $A C = \sqrt{5}$ as shown in the above figure:

Now consider $A B^{2} + B C^{2}$

$A B^{2} + B C^{2} = (\sqrt{2})^{2} + (\sqrt{3})^{2} = 2 + 3 = 5 = (\sqrt{5})^{2} = A C^{2}$

Since $A B^{2} + B C^{2} = A C^{2}$ , therefore we have verified Pythagoras theorem , $△ A B C$ is a right angled triangle.

Hence, $\sqrt{2}$ , $\sqrt{3}$ and $\sqrt{5}$ represent the sides of a right angled triangle.

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