ab=36cm m is midpoint of ab semi circle are drown on ab, am and mb as diameter a circle with centrec touches all the three circle find the area of shaded region 
Answers
Answer: 45 π cm²
Step-by-step explanation:
Let the centre of semicircles with diameter AM and MB be P and Q respectively and the circle with centre C touch the semicircles at E,F and R as shown.
Let radius of the circle with centre C be r cm.
AB = 36 cm and M is the mid -point of AB
AM = MB = 36/2 = 18cm...(1)
AP = PM = MQ =QB =18/2 cm = 9 cm...(2)
join RM. It passes through C and RM⊥ AB
MR= AM=18cm ...(3)
CM =RM - CR
= (18 - r)cm
PC = PE + CE
= (9+ r) cm
In right Δ CMP
PC² = CM² + PM²
(9+r)²= (18-r)² + 9²
∴ 81 +18r+ r² = 324 -36 r + r² + 81
18r + 36r = 324
54r = 324
r = 324/ 54
r = 6 cm
area of the shaded region
= ar( semicircle with centre m) - ar( semicircle with centre p) - (semicircle with centre Q) - ( semicircle with centre C )
=1/2 π (RM)²- 1/2 π (AP)²- 1/2 π (QB)²- πr²
= 1/2 π × 18² - 1/2 π ×9² - 1/2 π× 9²₋ π×6²
=1/2 × 324 π -1/2 (81π + 81π) - 36π
=162 π - 81π - 36 π
= 45π
