Questions: Which of the following states the Pythagorean theorem? A. In a right triangle, the hypotenuse equals the sum of the legs. B. In a right triangle, the hypotenuse equals the sum of the squares of the legs. C. In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. D. In a right triangle, the square of the hypotenuse equals the sum of the legs.

Which of the following states the Pythagorean theorem?
A. In a right triangle, the hypotenuse equals the sum of the legs.
B. In a right triangle, the hypotenuse equals the sum of the squares of the legs.
C. In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.
D. In a right triangle, the square of the hypotenuse equals the sum of the legs.

Transcript text: Which of the following states the Pythagorean theorem? A. In a right triangle, the hypotenuse equals the sum of the legs. B. In a right triangle, the hypotenuse equals the sum of the squares of the legs. C. In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. D. In a right triangle, the square of the hypotenuse equals the sum of the legs.

Solution

Solution Steps

Step 1: Understand the Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in trigonometry that applies to right triangles. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs).

Mathematically, this is expressed as: \[ c^2 = a^2 + b^2 \] where \(c\) is the hypotenuse, and \(a\) and \(b\) are the legs.

Step 2: Analyze the Options

Let's evaluate each option to see which one correctly states the Pythagorean theorem:

A. In a right triangle, the hypotenuse equals the sum of the legs.
This is incorrect because the hypotenuse is not equal to the sum of the legs; it is related to the squares of the legs.
B. In a right triangle, the hypotenuse equals the sum of the squares of the legs.
This is incorrect because the hypotenuse itself is not equal to the sum of the squares of the legs; it is the square of the hypotenuse that equals the sum of the squares of the legs.
C. In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.
This is correct because it matches the mathematical expression of the Pythagorean theorem.
D. In a right triangle, the square of the hypotenuse equals the sum of the legs.
This is incorrect because the square of the hypotenuse equals the sum of the squares of the legs, not the sum of the legs themselves.

Step 3: Determine the Correct Answer

Based on the analysis, the correct statement of the Pythagorean theorem is given in option C.