Circles
Chords and Intersecting Chords
Grade 11
Question:
<p>If two equal chords AB and CD of a circle intersect at point P, where ACB and CBD are minor segments of AB and CD, AP = 2 and PB = 7, then prove or disprove: PC = 7 and PD = 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: Equal chords intersecting inside a circle create symmetric segments: AP·PB = CP·PD, and for equal chords, AP = PD and PB = PC.
<p><strong>Analysis:</strong> For two equal chords AB and CD intersecting at P, by the intersecting chords theorem: \(AP \cdot PB = CP \cdot PD\). Since the chords are equal and symmetric about P, we have \(AP = PD\) and \(PB = PC\). Given \(AP = 2, PB = 7\), we get \(PC = 7, PD = 2\). Statement-1 is true and Statement-2 correctly explains this property.</p>
Correct Answer: A