Q6. - Geometry

Question

$\overline{RS}$ and $\overline{TU}$ are common internal tangents to the circles. If $RZ=4·7$ and $ZU=7·3$, Find $RS$ and $TU$.

Step-by-Step Solution

Answer

Value of, $RS=12$

And,        $TU=12$.

Step 1. Given information.

The figure:

$RZ=4.7$ And $ZU=7.3$

Step 2. Concept Used.

Theorem $9·1$ Corollary: 

Tangents to a circle from a point are congruent.

Step 3. Consider the given figure for further solution.

From the above figure it is clear that $\overline{RZ}$ and $\overline{TZ}$ are tangents to the smaller circle from the common point $Z$, so by theorem $9·1$ corollary, it can be said that:

$TZ=RZ=4.7$                                                                                 ….. (1)

Similarly, $\overline{UZ}$ and $\overline{SZ}$ are tangents to the bigger circle from the common point $Z$, so by theorem $9·1$ corollary, it can be said that:

$SZ=UZ=7.3$                                                                                ….. (2)

Step 4. Thus, from (1) and (2).

$\begin{array}{c}RS=RZ+ZS\\ =4.7+7.3\\ =12\end{array}$

And

$\begin{array}{c}TU=TZ+ZU\\ =4.7+7.3\\ =12\end{array}$

Therefore, Value of, $RS=12$, and $TU=12$.