Mathematics
In the given figure, AB is a diameter of the circle. Chords AC and AD produced meet the tangent to the circle at point B in points P and Q respectively. Prove that :
AB2 = AC × AP
Circles
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Answer
Join BC.
We know that,
The diameter of a circle subtends an angle of 90° at any point on circle.
∴ ∠ACB = 90°
We know that,
A tangent line is perpendicular to the radius line from the center to the point of contact
∴ ∠ABP = 90°
In △ACB and △ABP,
∠ACB = ∠ABP = 90°
∠A = ∠A [Common]
∴ △ACB ~ △ABP [By A.A. axiom]
We know that :
Ratio of corresponding sides of similar triangle are proportional.
⇒ AB2 = AC × AP
Hence, proved that AB2 = AC × AP.
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