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

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

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New post 28 Oct 2017, 23:01
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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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Re: If the circle below has centre O and length of the arc RST is 18pi  [#permalink]

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New post 28 Oct 2017, 23:24
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

to find the radius
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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New post 28 Oct 2017, 23:40
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.
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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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New post 04 Dec 2017, 04:16
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??
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If the circle below has centre O and length of the arc RST is 18pi  [#permalink]

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New post 04 Dec 2017, 07:06
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

Answer 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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New post 12 Jan 2018, 07:11
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.

Answer: 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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New post 12 Jan 2018, 12:59
Bhadri1199 wrote:
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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Re: If the circle below has centre O and length of the arc RST is 18pi  [#permalink]

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New post 13 Jan 2018, 13:39
Bhadri1199 wrote:
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.
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Re: If the circle below has centre O and length of the arc RST is 18pi   [#permalink] 13 Jan 2018, 13:39
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