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In the figure above, square \(ABCD\) has an area of \(25.\) What is the area of the circle with center \(O?\)

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In the figure above, square \(ABCD\) has an area of \(25.\) What is the area of the circle with center \(O?\)

by Gmat_mission » Thu Sep 10, 2020 1:06 am

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In the figure above, square \(ABCD\) has an area of \(25.\) What is the area of the circle with center \(O?\)

A. \(\dfrac{5\sqrt2\pi}2\)

B. \(\dfrac{25\pi}4\)

C. \(\dfrac{25\pi}2\)

D. \(5\sqrt2\pi\)

E. \(25\pi\)

Answer: C

Source: Princeton Review
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Re: In the figure above, square \(ABCD\) has an area of \(25.\) What is the area of the circle with center \(O?\)

by gentvenus » Thu Sep 10, 2020 4:33 am
I keep getting B for some reason...
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Re: In the figure above, square \(ABCD\) has an area of \(25.\) What is the area of the circle with center \(O?\)

by ravinikhil » Thu Sep 10, 2020 6:16 am
ABCD is a square with area 25.
now let side length be a.
hence a^2 = 25
ie a = 5.
Now, a= AD = 5.
Imagine drawing a line from O to X which is on AD and bisects it in half.
Now consider AOX triangle
here AX is 5/2 in length, since ABCD is a square , hence angle OAD will be 45 degrees.
Now OA cos 45 = AX. { understood right }
OA / sqrt ( 2 ) = 5/2
OA = 5 * sqrt (2) / 2

Area of the circle is pie radius^2.
radius is OA.
hence the overall area is = pie * 25 / 2
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