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 the 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. The sector enclosed by the two radii is 1/3 the area of the entire circle.
Now here's a drawing of the problem above:
Let c = circumference.
Since angle A is 60 degrees, the corresponding central angle is 120 degrees. 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:
2/3c = 24
c = 36.
Since c = πd, we get:
Ï€d = 36.
d ≈ 11.
The correct answer is
C.
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