Bunuel wrote:

Two identical circles of area 144π are shown above. If the distance from point A to point B is 12, what is the perimeter of the shaded region?

A. 4π

B. 6π

C. 8π

D. 12π

E. 24π

Attachment:

The attachment **image008.jpg** is no longer available

Attachment:

image008ed.png [ 32.15 KiB | Viewed 480 times ]
Perimeter of the shaded region =

arc length AB * 2 (i.e., not chord AB)

Identical circles have identical circumferences

Arc length = fraction of circumference

To find that fraction, we need the central angle for the sector whose arc is AB

(A sector's arc length is that portion of the circumference subtended by sector's central angle.)

1) Sector with arc AB? Draw a triangle in one of the circles

Draw two lines, from center O to A, and from O to B. OA and OB are radii

2) Side lengths OA and OB of ∆ OAB = radius length

Derive radius from area:

\(144π = πr^2\)

\(r^2 = 144\), \(r = 12\)3) ∆ OAB is equilateral; all sides have equal length

AB = OA = OB = 12, hence

All angles of ∆ OAB = 60°

Central angle of of

sector AOB = 60°

3) Sector AOB as a fraction of the circle?

\(\frac{Part}{Whole}=\frac{SectorAngle}{360}=\frac{60}{360}=\frac{1}{6}\)4) Length of arc AB? \(\frac{1}{6}\) of circumference

Circle's circumference, r = 12:

\(= 2πr = (2π * 12) = 24π\)Arc Length:

\((\frac{1}{6} * 24π) = 4π\)5) Perimeter of shaded region: (2 * length of arc AB) = \((2 * 4π) = 8π\)

Answer C

_________________

In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"