Mathematics
In the given figure, AB is a direct common tangent to two intersecting circles. Their common chord when produced intersects AB at P. Prove that P is the mid-point of AB.
Circles
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Answer
We know that,
If a chord and a tangent intersect externally, then the product of lengths of the segments of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection.
For circle 1:
∴ PA2 = PC × PD
For circle 2:
∴ PB2 = PC × PD
Thus,
PA2 = PB2
Taking square root on both sides,
PA = PB
Point P divides AB into two equal parts.
Hence, proved P is the mid-point of AB.
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