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

_________________

In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"