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# If the circle below has centre O and length of the arc RST is 18pi

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Intern
Joined: 29 Aug 2010
Posts: 9
If the circle below has centre O and length of the arc RST is 18pi  [#permalink]

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28 Oct 2017, 22:01
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55% (hard)

Question Stats:

63% (01:59) correct 37% (01:50) wrong based on 113 sessions

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If the circle below has centre O and length of the arc RST is 18 pi, what is the perimeter of the
region RSTU?

(A) 12 pi+ 18
(B) 12 pi+ 27
(C) 18 pi+ 27
(D) 18 pi+ 54
(E) 18 pi+ 108

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Joined: 02 Aug 2009
Posts: 8256
Re: If the circle below has centre O and length of the arc RST is 18pi  [#permalink]

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28 Oct 2017, 22:24
shibbirahamed wrote:
If the circle below has centre O and length of the arc RST is 18 pi, what is the perimeter of the
region RSTU?

(A) 12 pi+ 18
(B) 12 pi+ 27
(C) 18 pi+ 27
(D) 18 pi+ 54
(E) 18 pi+ 108

hi

length of arc RST = $$\frac{60}{360} * perimeter= \frac{2*pi*r}{6}=18*pi......r=54..$$
triangle is EQUILATERAL triangle as two sides are radius so RT = radius = 54

perimeter = 18*pi +54

D
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Re: If the circle below has centre O and length of the arc RST is 18pi  [#permalink]

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28 Oct 2017, 22:40
Length of the arc RST (which subtends $$60^{\circ}$$ at the center) = 2$$\pi$$rX$$\frac{60}{360}$$, where r is the radius
of the circle. Thus, we have:
2$$\pi$$rX$$\frac{60}{360}$$ = 2$$\pi$$r
=> r = 54

In triangle TOR, we have: RO = TO = radius of the circle

Hence, we have:
$$\angle$$OTR = $$\angle$$ORT = $$\frac{180^{\circ}-60^{\circ}}{2}$$ = $$60^{\circ}$$

Thus, triangle TOR is equilateral.
Thus, we have: RT = TO = RO = radius of the circle = 54.
Thus, perimeter of the region RSTU = (18$$\pi$$+54).
The correct answer is option D.
Intern
Joined: 24 Jun 2017
Posts: 5
Re: If the circle below has centre O and length of the arc RST is 18pi  [#permalink]

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04 Dec 2017, 03:16
shibbirahamed wrote:
Length of the arc RST (which subtends $$60^{\circ}$$ at the center) = 2$$\pi$$rX$$\frac{60}{360}$$, where r is the radius
of the circle. Thus, we have:
2$$\pi$$rX$$\frac{60}{360}$$ = 2$$\pi$$r
=> r = 54

In triangle TOR, we have: RO = TO = radius of the circle

Hence, we have:
$$\angle$$OTR = $$\angle$$ORT = $$\frac{180^{\circ}-60^{\circ}}{2}$$ = $$60^{\circ}$$

Thus, triangle TOR is equilateral.
Thus, we have: RT = TO = RO = radius of the circle = 54.
Thus, perimeter of the region RSTU = (18$$\pi$$+54).
The correct answer is option D.

how did u arrive that 2 angles would be equal??
Senior SC Moderator
Joined: 22 May 2016
Posts: 3703
If the circle below has centre O and length of the arc RST is 18pi  [#permalink]

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04 Dec 2017, 06:06
shibbirahamed wrote:
If the circle below has centre O and length of the arc RST is 18 pi, what is the perimeter of the
region RSTU?

(A) 12 pi+ 18
(B) 12 pi+ 27
(C) 18 pi+ 27
(D) 18 pi+ 54
(E) 18 pi+ 108

The sector is a fraction of the circle; the sector's central angle is a fraction of the circle's 360°.

Use that fraction, and the given arc length, to find circumference. Circumference yields radius, which we need for the sides of the triangle.

Sector as fraction of circle
$$\frac{Part}{Whole}=\frac{60}{360}=\frac{1}{6}$$

The sector is 1/6 of the circle. Its arc length is 1/6 of circumference.

Radius: Length of arc RST = 1/6 circumference
$$18π = 2πr * \frac{1}{6}$$
$$54πr = πr$$
, so $$r = 54$$

Triangle side lengths
Angles opposite equal sides are equal.
Sides OR and OT are radii: OR = OT
Triangle vertices at R and at T are angles opposite those equal sides.
So triangle vertex ∠R = vertex ∠T

One angle, at vertex O, is 60°
Let the other two equal angles = x
(2x + 60 = 180) --> (2x = 120)
x = 60
The triangle is equilateral (60°-60°-60°).
Two sides each = length of radius = 54
Third side, RT, must = 54

Perimeter of region RSTU
Arc RST + ∆ side RT
18π + 54

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Re: If the circle below has centre O and length of the arc RST is 18pi  [#permalink]

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12 Jan 2018, 06:11
shibbirahamed wrote:
If the circle below has centre O and length of the arc RST is 18 pi, what is the perimeter of the
region RSTU?

(A) 12 pi+ 18
(B) 12 pi+ 27
(C) 18 pi+ 27
(D) 18 pi+ 54
(E) 18 pi+ 108

Since OR and OT are the radii of the circle, we see that angle ORT = OTR = 60, so we have an equilateral triangle.

Arc RST corresponds to a 60-degree central angle, which is 60/360 = 1/6 of the circumference. If we let the circumference = n, we have:

(1/6)n = 18π

n = 108π

Since circumference = 2πr, the radius is 54, and thus each side of the equilateral triangle is 54.

Thus, the perimeter of region RSTU is 18π + 54.

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Senior SC Moderator
Joined: 22 May 2016
Posts: 3703
Re: If the circle below has centre O and length of the arc RST is 18pi  [#permalink]

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12 Jan 2018, 11:59
shibbirahamed wrote:
Length of the arc RST (which subtends $$60^{\circ}$$ at the center) = 2$$\pi$$rX$$\frac{60}{360}$$, where r is the radius
of the circle. Thus, we have:
2$$\pi$$rX$$\frac{60}{360}$$ = 2$$\pi$$r
=> r = 54

In triangle TOR, we have: RO = TO = radius of the circle

Hence, we have:
$$\angle$$OTR = $$\angle$$ORT = $$\frac{180^{\circ}-60^{\circ}}{2}$$ = $$60^{\circ}$$

Thus, triangle TOR is equilateral.
Thus, we have: RT = TO = RO = radius of the circle = 54.
Thus, perimeter of the region RSTU = (18$$\pi$$+54).
The correct answer is option D.

how did u arrive that 2 angles would be equal??

Bhadri1199 , I think maybe you did not track on a part of the post?
The two angles are equal because angles opposite equal sides are equal.
One unknown angle is opposite side OT. Side OT = radius
The other unknown angle is opposite side OR. Side OR also = radius
Two equal sides? Two equal angles opposite those sides.

That is the logic behind "Hence," shibbirahamed :
Quote:
In triangle TOR, we have: RO = TO = radius of the circle
Hence, we have:
$$\angle$$OTR = $$\angle$$ORT

Hope that helps.
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Intern
Joined: 30 Oct 2017
Posts: 8
Re: If the circle below has centre O and length of the arc RST is 18pi  [#permalink]

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13 Jan 2018, 12:39
shibbirahamed wrote:
Length of the arc RST (which subtends $$60^{\circ}$$ at the center) = 2$$\pi$$rX$$\frac{60}{360}$$, where r is the radius
of the circle. Thus, we have:
2$$\pi$$rX$$\frac{60}{360}$$ = 2$$\pi$$r
=> r = 54

In triangle TOR, we have: RO = TO = radius of the circle

Hence, we have:
$$\angle$$OTR = $$\angle$$ORT = $$\frac{180^{\circ}-60^{\circ}}{2}$$ = $$60^{\circ}$$

Thus, triangle TOR is equilateral.
Thus, we have: RT = TO = RO = radius of the circle = 54.
Thus, perimeter of the region RSTU = (18$$\pi$$+54).
The correct answer is option D.

how did u arrive that 2 angles would be equal??

First: you know that the radius is 54, and since the radius of a circle is 54 from the origin of O to any point on the circle is always 54 (r=54 for all angles from 1 to 360), then you know that those two sides are equal. And if those two sides are equal then you can assume the angles are equal.
Re: If the circle below has centre O and length of the arc RST is 18pi   [#permalink] 13 Jan 2018, 12:39
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