Question

which best explains whether a triangle with side lengths 5 cm, 13 cm, and 12 cm is a right triangle?

the triangle is a right triangle because (5^2 + 12^2 = 13^2).

the triangle is a right triangle because (5 + 13 > 12).

the triangle is not a right triangle because (5^2 + 13^2 > 12^2).

the triangle is not a right triangle because (5 + 12 > 13).

Explanation:

Step1: Recall Pythagorean theorem

For a right triangle, the sum of the squares of the two shorter sides (legs) equals the square of the longest side (hypotenuse), i.e., \(a^2 + b^2 = c^2\) where \(c\) is the hypotenuse.

Step2: Identify sides and apply theorem

The sides are 5, 12, 13. The longest side is 13. Calculate \(5^2 + 12^2\) and \(13^2\).
\(5^2 = 25\), \(12^2 = 144\), so \(5^2 + 12^2 = 25 + 144 = 169\).
\(13^2 = 169\). Thus, \(5^2 + 12^2 = 13^2\), satisfying the Pythagorean theorem.
Options with \(5 + 13 > 12\) or \(5 + 12 > 13\) refer to triangle inequality (for triangle existence), not right triangle criteria. The third option's calculation is incorrect as it uses the wrong sides.

Answer:

The triangle is a right triangle because \(5^2 + 12^2 = 13^2\).