Question

In the 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

seg OP is the radius.   ...[Given]

∴ 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]

∴ `QS = 1/2 xx 12`

∴ QS = 6 cm

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

∴ `OQ = 1/2 r`

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

∴ OS² = OQ² + QS²    ...[Pythagoras theorem]

∴ `r^2 = (1/2 r)^2 + 6^2`

∴ `r^2 = 1/4 r^2 + 36`

∴ `r^2 - 1/4 r^2 = 36`

∴ `3/4 r^2 = 36`

∴ `r^2 = (36 xx 4)/3`

∴ r² = 48

∴ r = `sqrt(48)`

∴ r = `4sqrt(3)` cm   ...[Taking square root of both sides]

∴ The radius of the given circle is `4sqrt(3)` cm.