Find the length of the tangent drawn to a circle of radius $8 c m$ , from a point which is at a distance of $10 c m$ from the centre of the circle.

3 c m

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18 c m

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12 c m

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6 c m

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Solution

The correct option is D $6 c m$
$G i v e n - O i s t h e c e n t r e o f a c i r c l e t o w h i c h a t a n g e n t P Q h a s b e e n d r a w n t o t h e c i r c l e a t Q w h e n O P = 10 c m . T h e r a d i u s o f t h e g i v e n c i r c l e = 8 c m . T o f i n d o u t - T h e l e n g t h o f t h e t a n g e n t P Q = ? S o l u t i o n - W e j o i n O Q . ∴ O q i s a r a d i u s o f t h e c i r c l e t h r o u g h t h e p o i n t o f c o n t a c t T o f t h e t a n g e n t P Q . W e k n o w t h a t t h e r a d i i t h r o u g h t h e p o i n t o f c o n t a c t o f a t a n g e n t t o a c i r c l e i s p e r p e n d i c u l a r t o t h e t a n g e n t . ∴ O Q ⊥ P Q ⟹ ∠ O Q P = 90^{o} . ∴ Δ O Q P i s a r i g h t o n e w i t h O P a s h y p o t e n u s e . S o, a p p l y i n g P y t h a g o r a s t h e o r e m, w e g e t P Q = \sqrt{{O P}^{2} - {O Q}^{2}} = \sqrt{10^{2} - 8^{2}} c m = 6 c m . ∴ T h e l e n g t h o f t h e t a n g e n t P Q = 6 c m A n s - O p t i o n - D$

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The radius of a circle is 8 cm. Calculate the length of a tangent drawn to this circle from a point at a distance of 10 cm from its centre.

Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct a pair of tangents to the circle and find their lengths.

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The length of the tangent drawn from a point $8 c m$ away from the centre of a circle of radius $6 c m$ is $2 \sqrt{7} c m .$

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