Question

Suppose that line l is tangent to a circle at a point P, and that secant m, which is parallel to l, intersects the circle at two points A and B. Prove that chords AP=BP

Answer 1

To prove that chords AP and BP are equal in a circle with a tangent line and a parallel secant line, the tangent line establishes point P as a midpoint, demonstrating that AP equals BP since tangents from a point are congruent.

Step-by-step explanation:

The student is asking to prove that in a circle where a tangent line l intersects at point P, and a secant line m, which is parallel to l, intersects the circle at points A and B, the chords AP are equal to BP.

Since line m is a secant and parallel to the tangent l, point P acts as a midpoint for the chord formed by A and B. This concludes that AP = BP because tangents from a point to a circle are congruent and the perpendicular from the tangent to the chord bisects the chord. Hence, it is proved that chords AP = BP.