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

Question

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

chetan2u · Answer

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

shibbirahamed · Answer

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.

Bhadri1199 · Answer

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

generis · Answer

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

ScottTargetTestPrep · Answer

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

generis · Answer

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  :
 
Hope that helps.

mekoner · Answer

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.

Fdambro294 · Answer

Rule: for any Isosceles Triangle, the height drawn from the Vertex in between the 2 equal sides and made perpendicular to the NON-Equal Side will be a “Line of Symmetry” for that Isosceles Triangle -

Height = Median = Angle Bisector = Perpendicular Bisector

For the inscribed triangle in the picture:

OR = OT = radius

Drawing an Altitude from the center O ——> perpendicular to the non-equal side RT will Bisect the 60 degree angle and create Two 30-60-90 Triangles

Since the 60 degree angles will be the Inscribed Angles, the Triangle is an Equilateral Triangle with each angle equal to 60 degrees.

Each side will be equal.

RT = OR = OT = Radius

And the perimeter of the region will be given by:

(Length of Minor Arc 18(pi)) + (Length of Radius)

18(pi) / (Circumference of Entire Circle) = (60 degree angle created at the Center by the Arc) / (360 degrees)

18(pi) / C = 60/360 = 1/6

C = 108(pi) = (2) (pi) (R)

Where R = radius of circle

R = 54

(D) 18(pi) + 54

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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

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