Question

In the adjoining figure, \[OA = 5\]cm,$AB = 8$cm and $OD$ is perpendicular to $AB$ then $CD$ will be equal to

A.$2$cm
B.$3$cm
C.$4$cm
D.$5$cm

Answer 1

Hint: The perpendicular line drawn from the center of the circle on any chord will bisect the chord which means that line will be the perpendicular bisector of the chord. So, by this we can find the value of $OC$ then by subtracting its value from $OD$ we will get the value of $CD$ .

Complete step by step solution:
First of all in these types of questions we start solving the question by listing the given quantities in the question.
So, here in the question it is given that-
A circle with center $O$ and radius $r$ is drawn,
Further it is given that a line $OD$ is drawn perpendicular to the line $AB$
And hence we can conclude that the line $OD$ will be the perpendicular bisector of the chord $AB$, because we know that
The perpendicular line drawn from the center of the circle on any chord will bisect the chord that is that line will be the perpendicular bisector of the chord.
And also it is given that length of \[OA = 5\]cm and length of $AB = 8$cm
So, here by observation we can see that the line $OA$ and $OD$ can also be called as radius.
So, length of $OA = OD = r$
And we know that length of \[OA = 5\]cm
$ \Rightarrow OA = OD = r = 5$ cm
And also it is given that length of the line $AB = 8$cm
And as we have concluded above that $OD$ will be the perpendicular bisector of the line $AB$
$ \Rightarrow $Length of $AC = $ Length of $CB = $ $\dfrac{1}{2}$ Length of $AB$
$ \Rightarrow AC = CB = \dfrac{8}{2} = 4$ cm
So, now by considering the $\vartriangle OAC$
We know that $AC = 4$ cm
$OA = 5$ cm
And we know that this triangle is a Right angled triangle because $OC$ is perpendicular to $AC$
$ \Rightarrow $By applying Pythagoras theorem on the $\vartriangle OAC$
We can say that $O{C^2} + A{C^2} = O{A^2}$
As this has been given to us that
$AC = 4$and $OA = 5$
So, by putting the corresponding values we can say that
$
  {4^2} + O{C^2} = {5^2} \\
   \Rightarrow O{C^2} = {5^2} - {4^2} \\
   \Rightarrow O{C^2} = 25 - 16 \\
   \Rightarrow O{C^2} = 9 \\
   \Rightarrow OC = \sqrt 9 \\
 $
$ \Rightarrow OC = 3$ cm
As we further that
$OC + CD = OD$
So, by putting corresponding values we can conclude that
$
  3 + CD = 5 \\
   \Rightarrow CD = 5 - 2 \\
 $
$ \Rightarrow CD = 2$ cm
Hence, Option A is the correct answer.

Note: In these types of questions students generally forgot that the perpendicular drawn from the center of the circle on any chord would bisect that chord. So, these types of properties of triangles inscribed in a circle should be remembered by the students.