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 RSTUArc RST + ∆ side RT
18π + 54
Answer D
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