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PQ, QR and PR are diameters of the three circles shown above. If QR =

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PQ, QR and PR are diameters of the three circles shown above. If QR = [#permalink]

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PQ, QR and PR are diameters of the three circles shown above. If QR = 4, and PQ = 2QR, what is the area of the shaded region?

(A) 6π
(B) 8π
(C) 10π
(D) 12π
(E) 16π

[Reveal] Spoiler:
Attachment:
2017-11-30_1001_002.png
2017-11-30_1001_002.png [ 8.33 KiB | Viewed 572 times ]
[Reveal] Spoiler: OA

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Re: PQ, QR and PR are diameters of the three circles shown above. If QR = [#permalink]

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New post 29 Nov 2017, 22:23
D.

From question we can calculate PR=12 PQ=8.
Now shaded area= 0.5*(area of circle of dia PR-area of circle of dia PQ)+0.5*(area of circle of dia QR)
On solving this gives pi/8*(12^2-8^2+4^2)
= pi/8*(96)
=12pi


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PQ, QR and PR are diameters of the three circles shown above. If QR = [#permalink]

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New post 30 Nov 2017, 07:20
Bunuel wrote:
Image
PQ, QR and PR are diameters of the three circles shown above. If QR = 4, and PQ = 2QR, what is the area of the shaded region?

(A) 6π
(B) 8π
(C) 10π
(D) 12π
(E) 16π

[Reveal] Spoiler:
Attachment:
The attachment 2017-11-30_1001_002.png is no longer available

Attachment:
eeeeee.png
eeeeee.png [ 13.05 KiB | Viewed 247 times ]

Find areas of circles, then divide by 2 (shaded and unshaded regions are in semicircles)
To find shaded area: Use just the bottom half + \(\frac{1}{2}\) Circle QR

Circle QR diameter = 4, r = 2
Circle QR area = \(4\pi\)
\(\frac{1}{2} * 4\pi = 2\pi\)
= \(\frac{1}{2}\) Area of Circle QR

Circle PQ diameter: (2*4) = 8, r = 4
Circle PQ area = \(16\pi\)
\(\frac{1}{2} * 16\pi = 8\pi\)
= \(\frac{1}{2}\) Area of Circle PQ

Circle PR diameter: (4 + 8 = 12), r = 6
Circle PR area = \(36\pi\)
\(\frac{1}{2} * 36\pi = 18\pi\)
= \(\frac{1}{2}\) Area of Circle PR

Shaded area = Use just the bottom half (\(\frac{1}{2}\) PR + \(\frac{1}{2}\) QR) - \(\frac{1}{2}\)PQ
\(\frac{1}{2}\) QR is the little shaded area in the top half

Shaded area = Areas of
(\(\frac{1}{2}\) of shaded large Circle PR + \(\frac{1}{2}\) shaded little Circle QR)* - \(\frac{1}{2}\) unshaded medium Circle PQ
\((18\pi + 2\pi) - 8\pi\)
\(20\pi - 8\pi = 12\pi\)


Answer D

No need to account for the \(\frac{1}{2}\) shaded little Circle QR in the bottom half; it is included in \(\frac{1}{2}\) big Circle QR in the bottom half

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Re: PQ, QR and PR are diameters of the three circles shown above. If QR = [#permalink]

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New post 30 Nov 2017, 11:22
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Expert's post
Bunuel wrote:
Image
PQ, QR and PR are diameters of the three circles shown above. If QR = 4, and PQ = 2QR, what is the area of the shaded region?

(A) 6π
(B) 8π
(C) 10π
(D) 12π
(E) 16π

[Reveal] Spoiler:
Attachment:
2017-11-30_1001_002.png


We know that \(QR = 4\).

Thus \(PQ = 2QR = 8\) and \(PR = PQ + QR = 8 + 4 =12\)

The area of the shaded region = Area of semi-circle with diameter QR + Area of semi-circle with diameter PR - Area of semi-circle with diameter PQ

Area of shaded region
\(= \frac{π}{2}*(\frac{4}{2})^2 + \frac{π}{2}*(\frac{12}{2})^2 - \frac{π}{2}*(\frac{8}{2})^2\)

\(=2π + 18π - 8π\)

\(= 12π\)


The correct answer is Option D.
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Re: PQ, QR and PR are diameters of the three circles shown above. If QR = [#permalink]

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New post 03 Dec 2017, 17:38
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Expert's post
Bunuel wrote:
Image
PQ, QR and PR are diameters of the three circles shown above. If QR = 4, and PQ = 2QR, what is the area of the shaded region?

(A) 6π
(B) 8π
(C) 10π
(D) 12π
(E) 16π

[Reveal] Spoiler:
Attachment:
2017-11-30_1001_002.png


We are given that QR = 4 and PQ = 2QR; thus, PQ = 8. To determine the area of the shaded region, we can first determine the area of semicircle PR and then subtract the area of semicircle PQ from it. Then we add in the area of semicircle QR. Notice that radii of semicircles PR, PQ and QR are 6, 4 and 2, respectively.

Area of semicircle PR = 6^2 x π x 1/2 = 18π

Area of semicircle PQ = 4^2 x π x 1/2 = 8π

Area of semicircle QR = 2^2 x π x 1/2 = 2π

Thus, the area of the shaded region is 18π - 8π + 2π = 12π.

Answer: D
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Kudos [?]: 1019 [1], given: 3

Re: PQ, QR and PR are diameters of the three circles shown above. If QR =   [#permalink] 03 Dec 2017, 17:38
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PQ, QR and PR are diameters of the three circles shown above. If QR =

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