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A is the center of the circle, and the length of AB
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01 Jun 2026, 11:14
01 Jun 2026, 11:14
Expert Reply
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Question Stats:
83% (02:14) correct
16% (02:52) wrong based on 12 sessions
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A is the center of the circle, and the length of AB is \(4 \sqrt{2}\). The blue shaded region is a square. What is the area of the shaded region?
GRE ab center of the circle.png [ 106.16 KiB | Viewed 450 times ]
\(\begin{aligned}
& A. 4(4-\pi) \\
& B. 4(8-\pi) \\
& C. 8(2-\pi) \\
& D. 8(8-\pi) \\
& E. 16(4-\pi)
\end{aligned}\)
Attachment:
GRE ab center of the circle.png [ 106.16 KiB | Viewed 450 times ]
\(\begin{aligned}
& A. 4(4-\pi) \\
& B. 4(8-\pi) \\
& C. 8(2-\pi) \\
& D. 8(8-\pi) \\
& E. 16(4-\pi)
\end{aligned}\)
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A is the center of the circle, and the length of AB
[#permalink]
05 Jun 2026, 06:38
05 Jun 2026, 06:38
1
Expert Reply
Explanation
Let \(r\) is the radius of the circle
then \(2r\) is the diameter of the circle or a side of the square.
Point A is the center of the circle, and B is the bottom-right corner of the square.
Using coordinates, place the square with corners (0, 0), (2r, 0), (2r, 2r), (0, 2r). Then
A=(r, r), B=(2r, 0).
Therefore,
\(AB=\sqrt{(2r−r)^2+(0−r)^2}=\sqrt{r^2+r^2}=r\sqrt{2}\)
\(AB=4\sqrt{2}\),
so
\(r\sqrt{2}=4\sqrt{2} ⇒r=4.\)
Thus the square's side length is
\(s=2r=8,\) and its area is \(s^2=8^2=64.\)
The circle's area is
\(πr^2=π(4^2)=16π.\)
The area of the shaded region is:
\(64-16π = 13.76\)
Only E equals 13.76
Answer: E
Let \(r\) is the radius of the circle
then \(2r\) is the diameter of the circle or a side of the square.
Point A is the center of the circle, and B is the bottom-right corner of the square.
Using coordinates, place the square with corners (0, 0), (2r, 0), (2r, 2r), (0, 2r). Then
A=(r, r), B=(2r, 0).
Therefore,
\(AB=\sqrt{(2r−r)^2+(0−r)^2}=\sqrt{r^2+r^2}=r\sqrt{2}\)
\(AB=4\sqrt{2}\),
so
\(r\sqrt{2}=4\sqrt{2} ⇒r=4.\)
Thus the square's side length is
\(s=2r=8,\) and its area is \(s^2=8^2=64.\)
The circle's area is
\(πr^2=π(4^2)=16π.\)
The area of the shaded region is:
\(64-16π = 13.76\)
Only E equals 13.76
Answer: E
gmatclubot
A is the center of the circle, and the length of AB [#permalink]
05 Jun 2026, 06:38
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