Chapter 9.7, Problem 12WE

To determine

To prove: $\overline{AB}\cong \overline{AC}$

Explanation of Solution

Given information:

Given figure,

$\overline{AB}$ is tangent to $\odot Q$ ;

$\overline{AC}$ is tangent to $\odot S$ .

Calculation:

When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment.

Fro point $Q$ ,

  $AD\cdot AE=A{B}^2\mathrm{..............}(1)$

For point $S$ ,

  $AF\cdot AG=A{C}^2$ ………(2)

When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment.

For point $R$ ,

  $AD\cdot AE=AF\cdot AG\mathrm{..........}(3)$

By using equation (1), (2) and (3)

  $A{B}^2=A{C}^2$

Take square root both sides as,

  $AB=AC$

Therefore, it is proved that $\overleftrightarrow{AB}\cong \overleftrightarrow{AC}$