Which of the following statements is true regarding the Pythagorean Theorem?

• A. In a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse: $a^2+b^2=c^2$.
• B. The Pythagorean Theorem applies to all triangles, regardless of their angles.
• C. In a right triangle, the sum of the lengths of the two legs is equal to the length of the hypotenuse: $a+b=c$.
• D. The Pythagorean Theorem states that $a^2-b^2=c^2$ for any triangle.

Answer: (A)

Verified step by step guidance

1. Recall the Pythagorean Theorem, which applies specifically to right triangles. It states that 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). This can be written as \(a^{2} + b^{2} = c^{2}\), where \(c\) is the hypotenuse.
2. Understand that this theorem does not apply to all triangles, only right triangles. For triangles without a right angle, this relationship does not hold.
3. Note that the theorem involves the squares of the side lengths, not just the side lengths themselves. So, the sum of the lengths of the legs is not equal to the length of the hypotenuse; rather, it is the sum of their squares that equals the square of the hypotenuse.
4. Recognize that the theorem does not involve subtraction of squares as in \(a^{2} - b^{2} = c^{2}\); this is not a correct statement of the Pythagorean Theorem.
5. Therefore, the true statement is: In a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse, expressed as \(a^{2} + b^{2} = c^{2}\).