Question

if from any point on the common chord of two intersecting circles tangents be drawn to the circles prove that they are equal

Answer 1

Answer:

Complete step by step answer:

We are given that tangents are drawn to two intersecting circles from a point that is drawn from their common chord. Let us illustrate this.



In the above figure, AB is the common chord, P is the point, PQ and PR are the tangents. We have to show PQ=PRPQ=PR .

Let us apply the Intersecting Secant-Tangent Theorem. Intersecting Secant-Tangent Theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment. From the above figure, let us consider tangent PQ and the secant PAB. Therefore, according to the Intersecting Secant-Tangent Theorem, we can write

⇒PQ2=PA×PB...(i)⇒PQ2=PA×PB...(i)

Similarly, for the tangent PR and secant PAB, we can write

⇒PR2=PA×PB...(ii)⇒PR2=PA×PB...(ii)

From equations (i) and (ii), we can equate the LHS since the RHS are similar.

PQ2=PR2⇒PQ2=PR2

Let us take the square root on both sides.

⇒PQ=PR⇒PQ=PR

Thus, tangents drawn from any point on the common chord of two intersecting circles are equal.

Hence proved.