In a triangle PQR, S is a point on the side QR such that PS is perpendicular to QR. Which of the following options is true?

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Accepted Answer

Given triangle PQR with point S on side QR such that PS is perpendicular to QR.

We want to find a relation involving the sides and segments.

Step 1: Since PS ⊥ QR, triangle PSQ and triangle PSR are right-angled at S.

Step 2: Apply the Pythagorean theorem in triangles PSQ and PSR:
- In triangle PSQ: PQ² = PS² + QS²
- In triangle PSR: PR² = PS² + SR²

Step 3: Adding these two equations:
PQ² + PR² = (PS² + QS²) + (PS² + SR²) = 2PS² + QS² + SR²

Step 4: Rearranging, we get:
PQ² + PR² - (QS² + SR²) = 2PS²

Step 5: Now, consider the option 2:
PR² + QS² = PQ² + SR²

Check if this holds:
From the Pythagorean relations, this is a known property derived from the right angle at S.

Hence, option 2 is correct.

Answer

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