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# If D is the midpoint of AC and arcs DF and DE are centered at C and A

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If D is the midpoint of AC and arcs DF and DE are centered at C and A  [#permalink]

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30 May 2015, 06:03
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If D is the midpoint of AC and arcs DF and DE are centered at C and A respectively, and If AB=BC=2, what is the area of the shaded region?

A. $$2-2pi$$
B. $$\frac{pi}{6}$$
C. 0.5
D. $$2-\frac{pi}{2}$$
E. $$4-pi$$

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Re: If D is the midpoint of AC and arcs DF and DE are centered at C and A  [#permalink]

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31 May 2015, 23:12
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Cool question that involves an isosceles triangle with partial circles. We are asked for the area of a very unusual shape inside of a familiar shape, so we are looking to find the areas of familiar shapes around it and then take the difference. The area of the triangle is easy to find, as we know the base and height are 2, so area = 1/2*b*h = 2. We can see the triangle is divided up into the weird shape that we are looking for, and then two partial circles. Because the triangle is an isosceles right triangle, the other two angles are each 45 degrees. Because of the centering of D in AC and of the arcs, we can see that the radius of each circle is just half of line AC. The 45-45-90 triangle ratio is x:x:x*sqrt2, so from that we know that AC = 2*sqrt2. Half of that, and the radii of the circles, is sqrt2. Using the central angle rule of circles, we can figure out that the area of each of these circle portions is 45/360 or 1/8 of the area of what the total circle would be, so together they are 1/4 the area of what one total circle would be (the two circle areas are equal in size). The area of the full circle would be pie*r^2 = pie*sqrt2^2 = 2*pie. Therefore, 1/4 of this area = 1/2pie. Taking the difference between the triangle and the two circle portions gives us the area of the shaded portion, 2 - 1/2*pie, or answer choice D.

I hope this helps!
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Re: If D is the midpoint of AC and arcs DF and DE are centered at C and A  [#permalink]

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01 Jun 2015, 01:52
reto wrote:
Attachment:
T7919.png

If D is the midpoint of AC and arcs DF and DE are centered at C and A respectively, and If AB=BC=2, what is the area of the shaded region?

A. $$2-2pi$$
B. $$\frac{pi}{6}$$
C. 0.5
D. $$2-\frac{pi}{2}$$
E. $$4-pi$$

Check other Shaded Region Problems in our Special Questions Directory.
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Re: If D is the midpoint of AC and arcs DF and DE are centered at C and A  [#permalink]

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01 Jun 2015, 04:06
Area of triangle = (1/2)*base*height=2
Radius of circles= r=AD=DC=AC/2=\sqrt{2}
Areaof each arc=pie*r^2/8

Hence area of shaded region = 2-pie/2
D
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Re: If D is the midpoint of AC and arcs DF and DE are centered at C and A   [#permalink] 01 Jun 2015, 04:06
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# If D is the midpoint of AC and arcs DF and DE are centered at C and A

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