Question

In the figure, O is the center of the circle. Find the length of CD, if AB=5cm.

Answer 1

Hint – In this question prove triangle AOB and COD similar using the SAS that is side angle side postulate. Then use the concept that the ratio of the respective sides must be equal that is $\dfrac{{CD}}{{AB}} = \dfrac{{OC}}{{OA}} = \dfrac{{OD}}{{OB}}$, this will help finding the value of side CD.

Complete step-by-step answer:
In the given circle center is O, and AB = 5cm
As we know that the line joining the center and any point on the circumference of the circle is called the radius of the circle.
So as we see that OA = OB = OC = OD = radius of the circle.
Now in the triangle AOB and COD.
OA = OC = radius of the circle
OB = OD = radius of the circle
$\angle AOB = \angle COD = {55^o}$ (given)
So by Side-angle-side (SAS) congruency
Both the triangles AOB and COD are similar.
So the ratio of the respective sides are equal
$ \Rightarrow \dfrac{{CD}}{{AB}} = \dfrac{{OC}}{{OA}} = \dfrac{{OD}}{{OB}}$
\[ \Rightarrow \dfrac{{CD}}{{AB}} = \dfrac{{OA}}{{OA}}\] [as OC = OA, radius of the circle]
Now substitute the value we have,
\[ \Rightarrow \dfrac{{CD}}{5} = 1\]
$ \Rightarrow CD = 5$ cm.
So this is the required answer.

Note – Understanding the diagrammatic representation of such types of questions is very helpful in solving problems of this kind. Other triangle similar postulates are AAA that is angle angle angle, SSS that is side side side, ASA that is angle side angle, SAS that is side angle side etc. Proving two triangles similarly helps establishing relationships between the other sides of the same triangle using the concept of CPCT that is congruent part of congruent triangles.