Which type of triangle is required for the Pythagorean Theorem to apply, as shown in the figure below?

• A. A scalene triangle with no right angle
• B. An equilateral triangle
• C. A $\mathrm{right}\mathrm{triangle}$
• D. An isosceles triangle that is not right

Answer: (C)

Verified step by step guidance

1. Recall the Pythagorean Theorem states that in a triangle, the square of the length of the hypotenuse equals the sum of the squares of the other two sides, expressed as \(a^{2} + b^{2} = c^{2}\).
2. Understand that the Pythagorean Theorem specifically applies only to right triangles, which are triangles that have one angle exactly equal to 90 degrees.
3. Recognize that other types of triangles, such as scalene triangles without a right angle, equilateral triangles, or isosceles triangles without a right angle, do not satisfy the conditions for the Pythagorean Theorem to hold true.
4. Therefore, identify that the triangle required for the Pythagorean Theorem to apply must be a right triangle.
5. Summarize that the key property enabling the Pythagorean Theorem is the presence of a right angle, which distinguishes right triangles from other triangle types.