Question

RT is a common tangent to circle P and circle Q. Points R and S are the points of tangencyPR and OS are radii of the given circlesQS = 7. ST = 24. RT = 48

Answer 1

Step 1

Given data

QS = 7. ST = 24. RT = 48

Step 2:

To find PR, apply the theorem of similar triangles.

`\begin{gathered} (PR)/(RT)\text{ = }(QS)/(ST) \\ (PR)/(48)\text{ = }(7)/(24) \\ 24PR\text{ = 48 }*\text{ 7} \\ PR\text{ = }\frac{48\text{ }*\text{ 7}}{24} \\ PR\text{ = 2 }*\text{ 7} \\ PR\text{ = 14} \end{gathered}`

Step 3

Apply the Pythagoras theorem to find QT

`\begin{gathered} QS^2\text{ + QT}^2\text{ = ST}^2 \\ 7^2\text{ + QT}^2\text{ = 24}^2 \\ QT^2\text{ = 576 - 49} \\ QT^2\text{ = 527} \\ QT\text{ = }√(527) \\ QT\text{ = 22.9564805665} \\ QT\text{ = 22.96} \end{gathered}`

Step 4

Apply the Pythagoras theorem to find PT

`\begin{gathered} PT^2+PR^2\text{ = RT}^2 \\ PT^2+\text{ 14}^2\text{ = 48}^2 \\ PT^2+196\text{ = 2304} \\ PT^2\text{ = 2304 - 196} \\ PT^2\text{ = 2108} \\ PT\text{ = }√(2108) \\ PT\text{ = 45.912961133} \\ PT\text{ = 45.9} \end{gathered}`