Using the Pythagorean Theorem: Reverse use of theorem

To determine if the given triangle is right-angled, we will use the Pythagorean Theorem:

Step 1: Identify the sides and assign potential hypotenuse:
Assign $AC = 5$ as the hypotenuse since it is the longest side.

Step 2: Apply the Pythagorean Theorem:
According to the theorem, if the triangle is right-angled, it should satisfy:

$AB^2 + BC^2 = AC^2$

Substituting the given side lengths:

$3^2 + 4^2 = 5^2$

Calculate each square:

$9 + 16 = 25$

$25 = 25$

The equation holds true, confirming that the triangle is indeed right-angled.

Therefore, the solution to the problem is Yes.

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 $\text{No}$.

To determine if the triangle is right-angled, we apply the Pythagorean Theorem as follows:

• Identify the sides: $a = 2$, $b = 4$, and $c = 5$ (longest side).
• Apply the Pythagorean Theorem: check if $2^2 + 4^2 = 5^2$.

Calculate:
$2^2 = 4$
$4^2 = 16$
Sum: $4 + 16 = 20$

Next, calculate $5^2 = 25$.

Compare: $20 \neq 25$.

Since $20 \neq 25$, the inequality confirms the triangle is not right-angled.

The triangle is not right-angled, thus the correct answer is No.