Problem Solving

Hi,

Need help answering the question in the attached screen shot.

Thanks

Equilateral triangle ABC is inscribed in a circle (points ABC are on the circle). IF the length of arc ABC is 24, what is the approximate diameter of the circle

A) 5
B) 8
C) 11
D) 15
E) 19

A CENTRAL ANGLE is formed by two radii.
An INSCRIBED ANGLE is formed by two chords.
When an inscribed angle and a central angle intercept the SAME ARC on a circle, the degree measurement of the inscribed angle is 1/2 the degree measurement of the central angle:



Circles display the following proportionality:

(Central Angle)/360 = (intercepted arc length)/circumference = (sector area)/(circle area)

Since 120/360 = 1/3, the intercepted arc in the circle above is 1/3 the circumference of the circle, and the sector enclosed by the two radii is 1/3 the area of the entire circle.

Onto the problem above:



Let c = circumference.
Since ∠A is 60º, the corresponding central angle = 120º.
Since 120/360 = 1/3, arc BC = (1/3)c.
Using similar logic, arc AB = (1/3)c.
Thus, arc ABC = (2/3)c.

Since arc ABC = 24, we get:
24 = 2/3c
c = 36.

Thus:
Ï€d = 36.
d ≈ 11.

The correct answer is C.