Refer to the figure I have attached.
Since line OA = Line AP = 1, A line drawn from A perpendicular to OP will bisect it.
Hence, RP = 0.5 cm.
Now, RP = 0.5 cm and AP = 1 cm
Let x be the angle formed between APR
Therefore, cos(x) = RP/AP = 0.5/1 = \(60^{\circ}\) = \(\pi/3\)
Note : An easier way to approach this is that since AO = OP = AP = 1 cm, Triangle AOP will be equilateral and will have all sides equal to \(60^{\circ}\). Thus angle APR will be equal to \(60^{\circ}\) or \(\pi/3\)
Now, arc OA = l*x = 1*\(\pi/3\) = \(\pi/3\)
Now, perimeter of the shaded region = perimeter of all 3 circles - 8*(length of arc OA)
Note: we have to compensate for 8 arcs which are similar in length to OA
Perimeter of shaded region = 3*(2\(\pi\)r) - 8*\(\pi/3\) = \(10(\pi/3)\)
Answer : C
Attachments
perimeter.png [ 27.18 KiB | Viewed 7374 times ]
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