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`

∴ QR = 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]

∴ r² = `(1/2 "r")^2 + 6^2`

∴ r²  = `1/4 "r"^2` + 36

∴ r²  - `1/4 "r"^2` = 36

∴ `3/4 "r"^2` = 36

∴ r² = `(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.