Question

Prove theorem $9.2$

Short answer

A tangent can touch a circle at only one point.

Step-by-step solution

Step 1. Statement.

A tangent can touch a circle at only one point.

Step 2. Draw the figure.

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 $l\perp QR$.

Step 3. Concept used.

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.