Chapter 9.2, Problem 22WE

To determine

To prove: a tangent can touch a circle at only one point.

Explanation of Solution

Consider a line in the plane of a circle perpendicular to a radius at its outer endpoint.

In the below figure line $l$ is the plane of circle with center $Q$ and $l\perp QR$ .

Suppose $l$ is not tangent to circle then $l$ is touching the circle at some other point $P$ , then $QP=QR$ .

This is because both are radius, and they both make same angle.

Since $l\perp QR$ then $l\perp QP$ also but $\Delta QPR$ cannot have two $90°$ angle.

Therefore, $l$ cannot touch the circle at two points, so line $l$ touch the circle at only $R$ . Therefore by definition of tangent line $l$ is tangent to circle.