Construct a tangent to a circle of radius $4$ $c m$ from a point on the concentric circle of radius $6$ $c m$ and measure its length. Also verify the measurement by actual calculation.

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Solution

$S t e p s o f C o n s t r u c t i o n :$
$1. W i t h$ $^{'} O^{'}$ $a s c e n t e r t w o c i r c l e s a r e c o n s t r u c t e d w i t h r a d i i$ $4 c m$ $a n d$ $6 c m$ .
$2. D r a w$ $O M$ $r a d i u s f o r s m a l l c i r c l e a n d d r a w n p e r p e n d i c u l a r a t$ $M$ $w h i c h m e e t s b i g c i r c l e a t$ $P$ $a n d$ $Q$ .
$3. N o w$ $P Q$ $i s t h e s p a c e t a n g e n t d r a w n f o r s m a l l c i r c l e .$
$T a n g e n t$ $P M Q = 9 c m$
$V e r i f i c a t i o n : I n$ $⊥ O M P, ∠ M = 90^{o}$
$∴ O M^{2} + P M^{2} = O P^{2}$
$(4)^{2} + P M^{2} = (6)^{2}$
$P M^{2} = (6)^{2} - (4)^{2} = 36 - 16$
$P M^{2} = 20$
$∴ P M = \sqrt{20} = 4.5 c m$
$S i m i l a r l y,$ $M Q = 4.5 c m$ .
$∴ P Q = P M + M Q = 4.5 + 4.5$
$∴ P Q = 9 c m$ .

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