PA and PB are tangents to the circle at A and B respectively and the chord BC is parallel to tangent PA. If AC = 6 cm, and length of the tangent AP is 9 cm, then what is the length of the chord BC?
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Answer / Dheerendra Singh Sirohi
Since the chord BC is parallel to the tangent at A (PA), angle ABC (angle subtended by the chord on the center of the circle) must be a right angle. This means that angle BAC is also a right angle. Using the properties of tangents and angles in a triangle inscribed in a circle, we know that angle APB is 90 degrees as well. Thus, triangles ABP and ACP are similar (since they share an angle of 90 degrees). Their sides can be related as follows:
AP / AB = CP / AC
9 cm / AB = CP / 6 cm
AB = 54 cm
Since we know that BC is a diameter, the length of BC (2r) is twice the radius. To find the radius, we can use any of the tangents and the circle's center (point A):
9² + r² = (6 + r)²
r² - 18r + 81 = 0
r = (18 ± √(18² - 4 * 81)) / 2
r ≈ 7.62 cm
So the length of chord BC is approximately 2 * r = 2 * 7.62 cm = 15.24 cm.
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