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27 Nov 2019, 01:56
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In the figure above, point B has an x-coordinate of 0.5 and a y-coordinate that is higher than any other point on the semi-circle. If AB lies on the line y=2x+1, what is the area of the shaded region, in square units?
A. \(\frac{2\pi}{3}\)
B. \(\pi-\frac{\sqrt{3}}{8}\)
C. \(\pi-1\)
D. \(2\pi-\frac{3}{2}\)
E. \(\frac{3\pi}{8}\)
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27 Nov 2019, 04:41
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Bunuel
In the figure above, point B has an x-coordinate of 0.5 and a y-coordinate that is higher than any other point on the semi-circle. If AB lies on the line y=2x+1, what is the area of the shaded region, in square units?
A. \(\frac{2\pi}{3}\)
B. \(\pi-\frac{\sqrt{3}}{8}\)
C. \(\pi-1\)
D. \(2\pi-\frac{3}{2}\)
E. \(\frac{3\pi}{8}\)
Let a perpendicular dropped from Point B meets x-axis at point C
--> ACB is a right angle triangle with ∠C = 90 deg
--> Area of Shaded region = 1/4th the area of circle - Area of the triangle ABC
Let the point B = (0.5, b) & A = (a, 0)
--> y = 2x + 1
--> b = 2*0.5 + 1 = 1 + 1 = 2
--> Point B = (0.5, 2)
For point A,
0 = 2*a + 1
--> 2a = -1
--> a = -0.5
--> Point A = (-0.5, 0)
So, Point C = (0.5, 0)
--> Radius of the circle = BC = 2 - 0 = 2 units
AC = 0.5 - (-0.5) = 1
Area of Shaded region = 1/4*π*\(2^2\) - 1/2*AC*BC = π - 1/2*1*2 = π - 1
IMO Option C
devavrat
Joined: 19 Feb 2019
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Concentration: Marketing, Statistics
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GMAT 1: 650 Q46 V34
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08 Dec 2019, 22:25
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Quote:
Let a perpendicular dropped from Point B meets x-axis at point C
--> ACB is a right angle triangle with ∠C = 90 deg
--> Area of Shaded region = 1/4th the area of circle - Area of the triangle ABC
--> ACB is a right angle triangle with ∠C = 90 deg
--> Area of Shaded region = 1/4th the area of circle - Area of the triangle ABC
Shouldn't it be Area of shaded region = Area of semi circle - [Area of 1/4th circle +Area of triangle]??
I did not understand how you got 1/4th area of circle - Area of traingle??
