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17 Jun 2020, 06:22
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C
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29% (02:25) correct
71%
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wrong
based on 31
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A semicircle is inscribed in a quarter circle as shown above. What fraction of the quarter circle is shaded red?
A. 1/3
B. 1/2
C. 2/3
D. 11/15
E. 3/4
Are You Up For the Challenge: 700 Level Questions
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17 Jun 2020, 10:44
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The tangent is perpendicular to the radius .
AFEG is a square.
Suppose the radius of the semicircle is 1. Hence AE = √2
Suppose radius of quarter circle is r
Now
(DE)*(r+EA) = EH*EI
\((r-√2)(r+√2)=1*1\)
\(r = √3\)
Area of quarter circle =\(\frac{ \pi*3}{4}\)
Area of semi circle = \(\frac{\pi*1}{2}\)
Ratio= 2/3
Bunuel
17 Jun 2020, 11:32
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Since the radius of outer quarter circle is tangent to the inner semi-circle, therefore, ABCD is a square.
=> BD = root(2)*r
triangle DBE is right angled.
=> DE^2 = DB^2 + BE^2
=> R^2 = 2*r^2 + r^2 = 3*r^2
=> R^2 = 3*r^2
Area of semi circle = 0.5*pi*r^2
Area of quarter circle = 0.25*pi*R^2 = 0.25*pi*3*r^2 = 0.75*pi*r^2
ratio of areas = 0.5/0.75 = 2/3
Ans: C
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