Straight Lines
Quadrilateral Properties
Grade 11
Question:
<p>Let the diagonals of a convex quadrilateral ABCD intersect at point P, and let a, b, c, d denote the lengths of sides AB, BC, CD, and DA respectively. Then: Diagonals of quadrilateral ABCD are perpendicular if and only if \(a^2 + c^2 = b^2 + d^2\).</p>
<p>(A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1</p>
<p>(B) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1</p>
<p>(C) Statement-1 is true, Statement-2 is false</p>
<p>(D) Statement-1 is false, Statement-2 is true</p>
Step-by-Step Solution
Key Concept: When diagonals are perpendicular, opposite sides of the quadrilateral satisfy \(a^2 + c^2 = b^2 + d^2\) by Pythagorean theorem applied to the four right triangles formed.
<p><strong>Analysis:</strong> For a convex quadrilateral with perpendicular diagonals intersecting at P, applying the Pythagorean theorem to each triangle formed by the diagonals: \(AB^2 = AP^2 + BP^2\), \(BC^2 = BP^2 + CP^2\), \(CD^2 = CP^2 + DP^2\), \(DA^2 = DP^2 + AP^2\). Summing: \(a^2 + c^2 = AP^2 + BP^2 + CP^2 + DP^2\) and \(b^2 + d^2 = BP^2 + CP^2 + DP^2 + AP^2\). Thus \(a^2 + c^2 = b^2 + d^2\). Statement-2 establishes the inequality framework that confirms this equality holds at the boundary condition.</p>
Correct Answer: A