Question

In Figure, $AB$ and $CD$ are common tangents to two circles of unequal radii. Prove that $AB=CD.$

Answer 1

Step 1: Construction.

According to the given details, $AB$and $CD$ are common tangents to two circles.

Extend $AB$and $CD,$to intersect at $P.$

Step 2: To prove $AB=CD.$

Consider the circle with a greater radius.

Tangents drawn from an external point to a circle are equal

$AP=CP...\left(i\right)$

Now, consider the circle with a smaller radius.

$BP=BD...\left(ii\right)$

On subtracting equation $\left(ii\right)$ from $\left(i\right)$, we get;

$$\begin{gathered} AP-BP=CP-BD \\ AB=CD \end{gathered}$$

Hence proved that $AB=CD.$