With the vertices of $△ P Q R$ as centres, three circles are described, each touching the other two externally. If the sides of the triangle are $7 c m, 8 c m$ and $11 c m$ . The radii of the three circles are

4 c m, 5 c m & 2 c m

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2 c m, 5 c m & 6 c m

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2 c m, 10 c m & 6 c m

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5 c m, 5 c m & 3 c m

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Solution

The correct option is B $2 c m, 5 c m & 6 c m$
$G i v e n - Δ P Q R h a s s i d e s P Q = 7 c m, Q R = 8 c m & R P = 11 c m . P, Q & R a r e t h e c e n t r e s o f t h r e e c i r c l e s t o u c h i n g e a c h o t h e r .$

$T o f i n d o u t - t h e r a d i i o f t h e g i v e n c i r c l e s .$

$S o l u t i o n -$

$L e t u s t a k e t h e r a d i u s o f t h e c i r c l e w i t h c e n t r e P = x c m, t h e r a d i u s o f t h e c i r c l e w i t h c e n t r e Q = y c m a n d t h e r a d i u s o f t h e c i r c l e w i t h c e n t r e R = z c m .$

$∴ x + y = 7..... (i), y + z = 8........ (i i) & z + x = 11...... (i i i)$

$s i n c e t h e d i s t a n c e b e t w e e n t h e c e n t r e s o f t w o c i r c l e s, t o u c h i n g e a c h o t h e r, i s t h e s u m o f t h e i r r a d i i .$

$A d d i n g (i), (i i) & (i i i) w e g e t x + y + z = \frac{1}{2} \times 26 = 13 c m . . . . . . (i v) .$

$S u b t r a c t i n g (i) f r o m (i v) w e h a v e z = (13 - 7) c m = 6 c m,$

$s u b t r a c t i n g (i i) f r o m (i v) w e h a v e x = (13 - 8) c m = 5 c m &$

$s u b t r a c t i n g (i i i) f r o m (i v) w e h a v e y = (13 - 11) c m = 2 c m .$

$S o t h e r a d i i a r e 5 c m, 2 c m & 6 c m . A n s - O p t i o n B .$

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With the vertices of △PQR as centres, three circles are described, each touching the other two externally. If the sides of the triangle are 7cm,8cm and 11cm. The radii of the three circles are

With the vertices of $△ P Q R$ as centres, three circles are described, each touching the other two externally. If the sides of the triangle are $7 c m, 8 c m$ and $11 c m$ . The radii of the three circles are