Question

A circle is circumscribed by the rhombus which in turn is made up by joining the mid-points of a rectangle whose sides are 12 cm and 16 cm respectively. What is the area of the circle? 

(a) \(\frac{625π}{26}\)

(b) \(\frac{676π}{25}\)  

(c) \(\frac{576π}{25}\) 

(d) Can’t be determined

Answer 1

Answer : (c) \(\frac{576\pi}{25}\)

Let ABCD be the given rectangle, PQRS, the rhombus obtained on joining the mid-points of ABCD and circle with centre O, the circle inscribed in the rhombus.

∴ DS = \(\frac{AD}{2}\) = 6 cm = RO 

DR = \(\frac{DC}{2}\) = 8 cm = OS

∴ In right angled triangle ROS, SR = \(\sqrt{{OS}^2 + {RO}^2}\)

= \(\sqrt{8^2 + 6^2}\)  = \(\sqrt{64 + 36}\) 

= \(\sqrt{100}\) 

= 10 cm

∴ Area of circle = πr² = π \(\times\) \(\big(\) \(\frac{48}{10}\big)\)² 

= \(\frac{576\pi}{25}\)