Question

In the given figure CB: QR and CA: QR and CA||PR. If \[AQ = 12cm\] , \[AR = 20cm\] and \[PB = CQ = 15cm\] , find PC and BR.

Answer 1

Hint:
Here, we have given the triangle PQR in which A, B and C is the point of the side QR, PR and PQ. Also, we have given lines AB and BC. We have to find the length of the PC and BR by using Basic proportionality Theorem.

Complete step by step solution:

In $\Delta PQR$, \[CA||PR\]
Here by applying basic proportionality Theorem,
 $ \Rightarrow \dfrac{{PC}}{{CQ}} = \dfrac{{RA}}{{AQ}} \to (1)$
Now, put the given values in the above equation
 $
   \Rightarrow \dfrac{{PC}}{{15}} = \dfrac{{20}}{{12}} \\
   \Rightarrow PC = \dfrac{{15 \times 20}}{{12}} \\
   \Rightarrow PC = 25cm \\
 $
Now, for the length of BR
In $\Delta PQR$ , $CB||QR$
Using basic proportionality theorem,
 $
   \Rightarrow \dfrac{{PC}}{{CQ}} = \dfrac{{PB}}{{BR}} \\
   \Rightarrow \dfrac{{25}}{{15}} = \dfrac{{15}}{{BR}} \\
   \Rightarrow BR = \dfrac{{15 \times 15}}{{25}} \\
   \Rightarrow BR = 9cm \\
 $

$ \Rightarrow BR = 9cm$ and \[ \Rightarrow PC = 25cm\]

Note:
Basic proportionality theorem states that the line drawn parallel to the one side of the triangle and intersecting the other two sides in distinct points then the other two sides are divided in the same ratio. Basic probability theorem states the sides are in the same ratio not the same length.