In Fig. 10.3, if OA = 5 cm, AB = 8 cm and OD is perpendicular to AB, then CD is equal to:


Given:

Radius of the circle = r = AO = 5 cm


Length of chord AB = 8 cm


Since the line drawn through the center of a circle to bisect a chord is perpendicular to the chord, therefore AOC is a right angled triangle with C as the bisector of AB.


∴ AC = 1/2(AB) = 8/2 = 4 cm


In right angled triangle AOC, by Pythagoras theorem, we have:


(AO)2 = (OC)2 + (AC)2


⇒ (5)2 = (OC)2 + (4)2


⇒ (OC)2 = (5)2 - (4)2


⇒ (OC)2 = 25 – 16


⇒ (OC)2 = 9


Take square root on both sides:


⇒ (OC) = 3


∴ The distance of AC from the center of the circle is 3 cm.


Now, OD is the radius of the circle, ∴ OD = 5 cm


CD = OD – OC


CD = 5 – 3


CD = 2


∴ CD = 2 cm

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