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.
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.
Related Questions
If AB and CD are two chords of a circle which when produced meet at a point P outside the circle such that PA = 12 cm, AB = 4 cm and CD = 10 cm, find PD.
In the given figure, two circles intersect each other at the points A and B. If PQ and PR are tangents to these circles from a point P on AB produced, show that PQ = PR.
In the given figure, PAT is tangent at A. If ∠ACB = 50°, find :
(i) ∠TAB
(ii) ∠ADB
In the given figure, PAT is tangent at A. If ∠TAB = 70° and ∠BAC = 45°, find ∠ABC.
