Chapter 12, Problem 14CR

To determine

To prove:Triangles $PAR$ and $SBR$ are similar.

Explanation of Solution

Given information:

  $\overline{AB }$ is a tangent with circle with center P.

  $\overline{RS }$ is a tangent with circle with center Q.

Proof:

  

It is given that,

  $\overline{AB }$ is a tangent with circle with center P.

  $\overline{RS }$ is a tangent with circle with center Q.

A diameter divides a circle into two semicircles.

RBS is a semicircle.

An angle inscribed in a semicircle is a right angle.

  $\angle B$ is a right angle.

A tangent line is perpendicular to the radius.

  $\overline{AP }\perp \overline{AR }$

Perpendicular lines form a right angle.

  $\angle PAR$ is a right angle.

Right angles are congruent.

  $\angle PAR\cong \angle B$

Vertical angles are congruent.

  $\angle ARP\cong \angle SRB$

Two triangles are similar under AA similarity rule.

  $\Delta PAR\sim \Delta SBR$