Question

$\left(a\right).$What do you think is true of common external tangents $\overline{AB}$and $\overline{CD}$$?$prove it.

$\left(b\right).$Will your results in part $\left(a\right)$be true if the circles are congruent$?$

Short answer

(a)

The tangent $AB$ is congruent to tangent $CD$.

Step-by-step solution

Step 1. Given information.

The figure here given is,

Step 2. Concept used.

In the given figure, $BZ$ and $DZ$ are tangents.

If circle has a two tangent from a unique point then there are equal,

$BZ=DZ$ and $AZ=CZ........\left(1\right)$

The sum of $BA$ and $AZ$ is $DZ$

Step 3. This implies as shown below.

$\begin{array}{l}BZ=DZ\\ BA+AZ=DC+CZ\end{array}$

It is known that $AZ=CZ$from (1)

Subtract $AZ$ from both sides to get,

$BA=DC$

Therefore, it is proved that $AB\cong CD$.

$\left(b\right)$

Yes, the results in part $\left(a\right)$ holds good if the circles are congruent.