<p>If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is __________ (up to three decimal places).</p>
Step-by-Step Solution
Key Concept: When two circles intersect at 90°, their tangents at the intersection point are perpendicular. Use the property that the line joining centers is perpendicular to the common chord, combined with the orthogonality condition to find the distance between centers.
<p><strong>Step 1:</strong> For two intersecting circles with radii r₁ = 5 cm and r₂ = 12 cm, when the angle of intersection is 90°, the circles are orthogonal.</p><p><strong>Step 2:</strong> For orthogonal circles, the condition is: d² = r₁² + r₂², where d is the distance between centers.</p><p>d² = 5² + 12² = 25 + 144 = 169</p><p>d = 13 cm</p><p><strong>Step 3:</strong> Let the common chord be AB, and M be its midpoint on line joining centers O₁ and O₂. If O₁M = x, then O₂M = (13 - x).</p><p><strong>Step 4:</strong> Using Pythagoras in triangles O₁MA and O₂MA:</p><p>AM² = 5² - x² = 25 - x²</p><p>AM² = 12² - (13-x)² = 144 - (169 - 26x + x²) = 26x - 25 - x²</p><p><strong>Step 5:</strong> Equating both expressions:</p><p>25 - x² = 26x - 25 - x²</p><p>50 = 26x</p><p>x = 25/13</p><p><strong>Step 6:</strong> Length of common chord = 2·AM = 2√(25 - (25/13)²)</p><p>= 2√(25 - 625/169) = 2√((4225 - 625)/169) = 2√(3600/169)</p><p>= 2 × (60/13) = 120/13 ≈ 9.231 cm</p><p>∴ Answer: <strong>9.231</strong></p>
Correct Answer: 9