Consider the region formed by the intersections of three semi circles whose diameters, of lengths 3, 4 and 5, from a triangle. What is the area of shaded region bounded by the arcs AB, BC and AC?
:
A
A Radius of semicircles ACB, AC and BC are 2.5, 2 and 1.5 respectively.
Area of Semi circles (ACB), (AC) and (BC) are 6.25π2, 2π and 2.25π2 respectively. And area of
triangle ABC =12 × 4 × 3 = 6
Sum of the area of region A and B = (Area of semicircle (ACB) - Area of triangle (ABC)} = (6.25π2−6)
Area of Shaded region = (Sum of the area of semicircles AC and BC - Sum of the area of region A and B)
Area of Shaded region = (2π+2.25π2) - (6.25π2−6) = (6.25π2) - (6.25π2−6) = 6
Alternate Solution:
As the options are far apart we can easily arrive at the answer by approximation. The shaded
region is approximately half of the areas of semicircles AC and BC = 14(4π+2.25π) = 6(approximately)
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