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प्रश्न
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.
उत्तर
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.
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