Question

In the adjoining figure, line l touches the circle with center O at point P, Q is the midpoint of radius OP. RS is a chord through Q such that chords RS || line l. If RS = 12, find the radius of the circle.

Answer 1

Let the radius of the circle be r. line l is the tangent to the circle and [Given] 

seg OP is the radius. 

∴ seg OP ⊥ line l [Tangent theorem] chord RS || line l [Given] 

∴ seg OP ⊥ chord RS 

∴ QS = 1/2 RS [Perpendicular drawn from the center of the circle to the chord bisects the chord] 

= 1/2 × 12 = 6 cm 

Also, OQ = 1/2 OP [Q is the midpoint of OP] 

= 1/2 r

In ∆OQS, ∠OQS = 90° [seg OP ⊥ chord RS ]

∴ OS² = OQ² + QS² [Pythagoras theorem] 

∴ r² = ( 1/2 r)² + 6² 

∴ r² = 1/4 r² + 36 

∴ 3/4 r²  = 36 

∴ r² = (36 x 4)/3

∴ r² = 48

∴ r = √48[Taking square root of both sides]

= 4√3 

∴ The radius of the given circle is 4√3 cm.