Verifying a Right Triangle: Is a Triangle with Sides 6, 6, and 12 Right-Angled?

To determine if the triangle with sides $AB = 6$, $BC = 12$, and $CA = 6$ is a right triangle, we will apply the reverse Pythagorean Theorem.

According to the Pythagorean Theorem, in a right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Therefore, we check if:

• $12^2 = 6^2 + 6^2$

Calculating, we have:

$12^2 = 144$
$6^2 = 36$
Thus, $6^2 + 6^2 = 36 + 36 = 72$

Since $144 \neq 72$, the condition $c^2 = a^2 + b^2$ fails to hold for any permutation of the given side lengths, indicating that none of the angles in the triangle is a right angle. Therefore, the triangle is not a right-angled triangle.

Therefore, the answer to the problem is $\text{No}$.