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# In the circle above, AC and BD are the diameters of length 10. The sum

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Math Expert
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In the circle above, AC and BD are the diameters of length 10. The sum  [#permalink]

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23 Nov 2017, 01:09
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Difficulty:

35% (medium)

Question Stats:

76% (01:44) correct 24% (01:21) wrong based on 53 sessions

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In the circle above, AC and BD are the diameters of length 10. The sum of the lengths of the arcs AXD and BYC is

(A) 4π
(B) 8π
(C) 9π
(D) 10π
(E) 16π

Attachment:

2017-11-22_1023_001.png [ 8.41 KiB | Viewed 765 times ]

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Re: In the circle above, AC and BD are the diameters of length 10. The sum  [#permalink]

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23 Nov 2017, 03:46
2

AXD and BYC are both angles which are supplementary to 36 degree, Therefore, AXD = BYC = 144
The diameter AC = BD = 10, which makes the radius(r) 5.

Given data: r=5 and θ=144

The length of the arc AXD is $$(\frac{θ}{360}) × 2π × r$$
Therefore, the length of the arc AXD is $$\frac{144}{360} * 2π * 5= \frac{4}{10} * 2π * 5= 4π$$

Since both the arcs have the exact same length , the total length of the arcs is 2*4π = 8π(Option B)
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Re: In the circle above, AC and BD are the diameters of length 10. The sum  [#permalink]

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23 Nov 2017, 08:05
1
Bunuel wrote:

In the circle above, AC and BD are the diameters of length 10. The sum of the lengths of the arcs AXD and BYC is

(A) 4π
(B) 8π
(C) 9π
(D) 10π
(E) 16π

Attachment:
2017-11-22_1023_001.png

The key is to find the areas of sectors AXD and BYC, combined, as a fraction of the circle.

That fraction will hold for arc lengths and circumference. Arc lengths of arcs AXD and BYC will be the same fraction of the circumference as the sectors are of the circle's area.

Central angles of arcs AXD and BYC

The smaller sectors' central angles are 36° each (they are vertical angles, hence equal).

Each small sector's central angle lies on a straight line with one larger sector's central angle. Larger sector AXD's central angle, e.g., therefore is

180° - 36° = 144°

Sectors AXD and BYC have vertical central angles. Both sectors' central angles hence = 144°. Together, their central angles total 288°.

Central angles determine fractional area. In other words, the central angles, summed, determine what fraction of the circle the sectors AXD and BYC are.

Sectors AXD and BYC make up what fraction of the circle?

$$\frac{Sectors'Angles}{360°}=\frac{Sectors'Area}{CircleArea}$$

$$\frac{288}{360}=\frac{4}{5}=\frac{Part}{Whole}$$

Sectors AXD and BYC make up $$\frac{4}{5}$$ of the circle. Their arc lengths will be $$\frac{4}{5}$$ of the circumference.

Radius of circle with diameter of 10:
$$10 = 2r$$, $$r = 5$$

Circumference: $$2πr = 10π$$

Arc lengths AXD and BYC are $$\frac{4}{5}$$ of circumference:

$$\frac{4}{5} * 10π = 8π$$

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Joined: 24 Aug 2017
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Re: In the circle above, AC and BD are the diameters of length 10. The sum  [#permalink]

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01 Dec 2017, 12:32
1
I got B as well. Can someone pls explain the answer?

Posted from my mobile device
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Joined: 02 Sep 2009
Posts: 49968
Re: In the circle above, AC and BD are the diameters of length 10. The sum  [#permalink]

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01 Dec 2017, 12:41
1
najsh2 wrote:
I got B as well. Can someone pls explain the answer?

Posted from my mobile device

The correct answer is B. Edited. Thank you.
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Re: In the circle above, AC and BD are the diameters of length 10. The sum &nbs [#permalink] 01 Dec 2017, 12:41
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