Which triangle property is necessary for applying the Pythagorean theorem?

Which triangle property is necessary for applying the Pythagorean theorem?

• A. The triangle is right-angled
• B. The triangle has all equal sides
• C. The triangle has three obtuse angles
• D. The triangle has two equal sides

Answer: (A)

Pythagorean theorem applies only when there is a right angle. In a right triangle, the two legs form the right angle and the side opposite that angle is the hypotenuse—the longest side. The theorem says the square of the hypotenuse equals the sum of the squares of the other two sides, which is a specific property of right triangles. If a triangle isn’t right-angled, that exact relationship doesn’t hold; you’d need a different formula, like the Law of Cosines, to relate the side lengths. The other options describe triangle properties that don’t guarantee this relationship: equilateral means all sides (and all angles) are equal, which isn’t about the Pythagorean setup; three obtuse angles can’t occur in a triangle; two equal sides is an isosceles property, not a condition for the Pythagorean theorem.

Pythagorean theorem applies only when there is a right angle. In a right triangle, the two legs form the right angle and the side opposite that angle is the hypotenuse—the longest side. The theorem says the square of the hypotenuse equals the sum of the squares of the other two sides, which is a specific property of right triangles. If a triangle isn’t right-angled, that exact relationship doesn’t hold; you’d need a different formula, like the Law of Cosines, to relate the side lengths.

The other options describe triangle properties that don’t guarantee this relationship: equilateral means all sides (and all angles) are equal, which isn’t about the Pythagorean setup; three obtuse angles can’t occur in a triangle; two equal sides is an isosceles property, not a condition for the Pythagorean theorem.