Question

$\overline{SR}$ is tangent to $\odot P$ and $\odot Q$.

$QT=6,TR=8,PR=30.$

$PQ=?,PS=?,ST=?$

Short answer

The required values are,

$PQ=20,PS=18\text{and}ST=16$

Step-by-step solution

Step 1. Given information.

$\odot P$Any $\overline{SR}$ is tangent to $\odot P$ and $\odot Q$. Also $QT=6,TR=8\text{and}PR=30$.

Step 2. Concept used.

Applying Pythagoras in triangle $\Delta RTQ$ to find the value of $PQ$.

Here, $\Delta RTQ$is right angle triangle,

$\begin{array}{l}Q{R}^2=6^2+8^2\\ Q{R}^2=36+64\\ QR=10\end{array}$

As value of ,

$\begin{array}{l}PQ=30-10\\ PQ=20\end{array}$

Step 3. Let ST be x and SP be y.

Here, $\overline{TQ}\left|\right|\overline{SP}$

Then,

$\begin{array}{l}\frac{8}{10}=\frac{x}{20}\\ x=\frac{8\cdot 20}{10}\\ x=16\end{array}$

Same way,

$\begin{array}{l}\frac{8}{6}=\frac{24}{y}\\ y=\frac{24\cdot 6}{8}\\ y=18\end{array}$

Therefore, the values are, $PQ=20,PS=18\text{and}ST=16$.