Question

In the adjoining figure, $AP = 3$ cm, $BA = 5$ cm and $CP = 2$ cm, find$CD$.

A) $12$ cm
B) $10$ cm
C) $9$ cm
D) $6$ cm

Answer 1

Hint: Try to prove the similarity of the two triangles and then solve this question using properties of similar triangles.

Complete step by step solution:
Here in this question we are given that,
$AP = 3$ cm, $BA = 5$ cm and $CP = 2$ cm
And here we can see that in the diagram that $PAB$and $PCD$ are two secants starting from a point $P$ which lies outside the circle.
And these two secants i.e $PAB$ and $PCD$ intersect the circle at $A$,$B$,$C$and $D$.
Here by observation we can see that the line $AC$ is parallel to the line $BD$.
So, firstly we will consider two triangles i.e $\vartriangle PAC$ and $\vartriangle PBD$
Here in these triangles we can see that,
$\angle APC = \angle BPD$ {Common angle of both triangles}………(i)
$\angle PCA = \angle PDB$ {Corresponding angles of parallel lines $AC$ and $BD$}………(ii)
Also, $\angle PAC = \angle PBD$ {Corresponding angles of parallel lines $AC$ and $BD$}………(iii)
So, by these results we can conclude that
$\vartriangle PAC \sim \vartriangle PBD$ by AAA(Angle Angle Angle) Rule.
$ \Rightarrow $ By properties of similar triangles we can say that
$\dfrac{{AP}}{{PC}} = \dfrac{{PD}}{{PB}}$ ……….(iv)
As it is given in the question that,
$AP = 3$ cm, $BA = 5$ cm and $CP = 2$ cm
$ \Rightarrow $Now considering equation (iv)
 $\dfrac{{AP}}{{PC}} = \dfrac{{PD}}{{PB}} = \dfrac{{PC + CD}}{{AB + AP}}$
$ \Rightarrow \dfrac{{AP}}{{PC}} = \dfrac{{PC + CD}}{{AB + AP}}$ ……..{As $PB = AB + AP$ and $PD = PC + CD$}
Now, by putting the corresponding values in the above equation we get,
$ \Rightarrow \dfrac{3}{2} = \dfrac{{2 + CD}}{{5 + 3}}$
$
   \Rightarrow 3\left( 8 \right) = 2\left( {2 + CD} \right) \\
   \Rightarrow 24 = 4 + 2\left( {CD} \right) \\
   \Rightarrow 20 = 2\left( {CD} \right) \\
   \Rightarrow 10 = CD \\
 $
$ \Rightarrow CD = 10$cm

$ \Rightarrow $ Option B is the correct answer.

Note:
This question can also be done by another method which is as follows,
 As we know that when two secants intersect the circle then
At the point of their intersection with each other i.e $P$, we have :
$PA \times PB = PC \times PD$
And here by putting the given values of lines in this formula, we can derive the correct answer