Verify whether the following measures represent the sides of a right angled triangle.
$6 \sqrt{3}, 12, 6$

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Solution

Let $△ A B C$ be a triangle where $A B = 6 \sqrt{3}$ , $B C = 6$ and $A C = 12$

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

$A B^{2} + B C^{2} = (6 \sqrt{3})^{2} + (6)^{2} = (36 \times 3) + 36 = 108 + 36 = 144 = (\sqrt{144})^{2} = 12^{2} = A C^{2}$

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, $6 \sqrt{3}$ , $6$ and $12$ represent the sides of a right angled triangle.

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