In the figure below, the rectangle at the corner measures $10 c m \times 20 c m .$ The corner A of the rectangle is also a point on the circumference of the circle. What is the radius of the circle in cm?

10 cm

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40 cm

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50 cm

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30 cm

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Solution

The correct option is C 50 cm
$W e t a k e t h e c o r n e r O o f t h e g i v e n s q u a r e O P Q R, c o n t a i n i n g t h e 10 c m \times 20 c m r e c t a n g l e, a s t h e o r i g i n e a n d t h e s i d e s O P & O R a s t h e c o o r d i n a t e a x e s . L e t t h e s i d e s o f O P Q R b e 2 a . S o O P Q R l i e s i n t h e t h i r d q u a d r a n t . T h e i n s c r i b e c i r c l e^{'} s r a d i u s i s r = a & c e n t r e i s C (a, a) . T h e c i r c l e p a s s e s t h r o u g h A (- 10, - 20) . ∴ T h e e q u a t i o n o f t h e c i r c l e i s {(x - h)}^{2} + {(y - k)}^{2} = r^{2} . . . . . . . . . . (i) H e r e (h, k) = (a, a) a n d r = a ∴ e q u a t i o n (i) b e c o m e s {(x - a)}^{2} + {(y - a)}^{2} = a^{2} . . . . . . . . . . (i i) T h e c i r c l e p a s s e s t h r o u g h A (- 10, - 20) . i . e (x, y) = (- 10, - 20) S o e q u a t i o n (i i) r e d u c e s t o {(- 10 - a)}^{2} + {(- 20 - a)}^{2} = a^{2} . O n s o l v i n g f o r a w e g e t | a | = 50 u n i t s a n d 10 u n i t s . (a i s a l e n g t h . S o w e d o n o t c o n s i d e r t h e n e g a t i v e s i g n) . A l s o A (- 10, - 20) l i e o n t h e c i r c l e . m i d w a y b e t w e e n (a, 0) & (0, a) . S o a > 10 u n i t s & a > 20 u n i t s . ∴ w e r e j e c t a = 10. s o a = 50 u n i t s A n s - O p t i o n C$

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In the figure below, the rectangle at the corner measures 10cm×20cm. The corner A of the rectangle is also a point on the circumference of the circle. What is the radius of the circle in cm?

In the figure below, the rectangle at the corner measures $10 c m \times 20 c m .$ The corner A of the rectangle is also a point on the circumference of the circle. What is the radius of the circle in cm?