Question

A chord of length $ 20cm $ is drawn at a distance of $ 24cm $ from the centre of a circle. Find the radius of the circle.

Answer 1

Hint: As we know that a chord of a circle is a straight line segment whose endpoints both lie on the circle. We will draw a circle and a chord, then we will draw the perpendicular from the radius to the chord. It will take the form of the triangle which is a right angled triangle. Then we will use the Pythagoras theorem in the triangle and find the radius of the circle.

Complete step-by-step answer:
As per the question we have length $ 20cm $ and distance $ 24 $ cm. Let us draw a circle of centre $ O $ , radius $ OA,OC $ and chord $ AB $ .

Here in the given figure $ AB $ is the chord of length $ 20cm $ and $ OC $ is the distance of $ 24cm $ . We know that line drawn from the centre of the circle to the chord is perpendicular to the chord and also the bisector of the chord. So triangle $ AOC $ is a right angled triangle with right angled at $ C $ .
Also $ OC $ bisects $ AB $ .
So $ AC = CB = \ dfrac{{AB}}{2} $ . Therefore $ AC = BC = 10\;cm $ .
Now we have to use the Pythagoras theorem to find the value of $ OA $ .
Here $ AC $ is the base, $ OA $ is the hypotenuse and $ OC $ is the height of the triangle. By applying the Pythagoras theorem we get:
  $ 0A = \sqrt {(A{C^2} + O{C^2})} $ .
By substituting the values we get
 $ OA = \sqrt {({{10}^2} + {{24}^2})} = \sqrt {676} $ .
This gives the value of $ OA = 26\;cm $ .
Hence the length of the radius of the circle is $ 26\;cm $ .
So, the correct answer is “ $ 26\;cm $ .”.

Note: In this type of question, we should first draw the diagram according to the question and then solve further. Also we should know that perpendicular drawn from the centre of the circle to the chord bisects the chord. We should always use the Pythagoras theorem to find the unknown values in the triangle drawn.