If two tangents inclined at an angle $60^{0}$ are drawn to a circle of radius 3 cm. The distance of the chord joining the point of contact of the tangents to the circle is $1.5 c m .$ Then length of each tangent is equal to

\frac{3}{2} \sqrt{3}

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6 cm

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3 cm

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3 \sqrt{3}

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Solution

The correct option is D $3 \sqrt{3}$ cm
$G i v e n - P Q & P R a r e t a n g e n t s t o t h e c i r c l e w i t h c e n t e O . O N i s t h e d i s t a n c e o f O f r o m t h e c h o r d Q R . ∠ Q P R = 60^{O} T o f i n d o u t - P Q o r P R . S o l u t i o n - P Q & P R a r e t a n g e n t s t o t h e c i r c l e w i t h c e n t e O . ∴ P Q = P R . . . . (i) a n d ∠ O Q P = ∠ O R P = 90^{o} . . . . . (i i) ∴ ∠ O Q P + ∠ O R P + ∠ P = 90^{o} + 90^{o} + 60^{o} = 240^{o} (f r o m i i) ⟹ ∠ R O Q = 360^{o} - 240^{o} = 120^{o} . . . . . (i i i) (a n g l e s u m p r o p e r t y o f q u a d r i l a t e r a l s) N o w i n Δ O Q R O Q = O R (r a d i i o f t h e s a m e c i r c l e) ∴ Δ O Q R i s i s o s c e l e s . ⟹ ∠ O Q R = ∠ O R Q ∴ ∠ O Q R + ∠ O R Q = 2 ∠ O Q R . ∴ 2 ∠ O Q R + ∠ R O Q = 2 ∠ O Q R + 120^{o} = 180^{o} (f r o m i) & (a n g l e s u m p r o p e r t y o f t r i a n g l e s) ⟹ ∠ O Q R = ∠ R O Q = 30^{o} ∴ ∠ P Q R = ∠ O Q P - ∠ O Q R = 90^{o} - 30^{o} = 60^{o} ⟹ ∠ Q R P = 180^{o} - 60^{o} - 60^{o} = 60^{o} ∴ Δ P Q R i s e q u i l a t e r a l i . e P Q = Q R . . . . (v) N o w O N i s t h e d i s t a n c e o f O f r o m t h e c h o r d Q R . ∴ O N ⊥ Q R o r Q N ⟹ Δ O Q N i s r i g h t a n g l e d a t N . S o, b y P y t h a g o r a s t h e o r e m, Q N = \sqrt{{O Q}^{2} - {O N}^{2}} = \sqrt{3^{2} - {1.5}^{2}} c m = 1.5 \sqrt{3} c m ⟹ Q R = 2 \times 1.5 \sqrt{3} c m = 3 \sqrt{3} c m . ∴ F r o m (v) P Q = 3 \sqrt{3} c m A n s O p t i o n D$

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Similar questions

Q. If two tangents inclined at an angle of

60^{\circ}

, are drawn to a circle of radius 3 cm, then length of each tangent (in cm) is equal to :

Q. If two tangents inclined at an angle of 60⁰ are drawn to a circle of radius 3 cm , then length of each tangent is equal to
(a)

\frac{3 \sqrt{3}}{2}

(b) 6 cm (c) 3 cm (d)

3 \sqrt{3}

If two tangents inclined at an angle $60^{\circ}$ are drawn to a circle of radius $3 c m,$ then the length of each tangent is
(A) $\frac{3}{2}$ $\sqrt{3} c m$
(B) $6 c m$
(C) $3 c m$
(D) 3 $\sqrt{3} c m$

Q. Question 9
If two tangents inclined at an angle

60^{\circ}

are drawn to a circle of radius 3 cm, then the length of each tangent is
(A)

\frac{3}{2}

\sqrt{3}

cm
(B) 6 cm
(C) 3 cm
(D) 3

\sqrt{3}

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If two tangents inclined at an angle 600 are drawn to a circle of radius 3 cm. The distance of the chord joining the point of contact of the tangents to the circle is 1.5cm. Then length of each tangent is equal to

If two tangents inclined at an angle $60^{0}$ are drawn to a circle of radius 3 cm. The distance of the chord joining the point of contact of the tangents to the circle is $1.5 c m .$ Then length of each tangent is equal to