Problem Solving

Hello,

I have a question here. So if the arc is 24 units and the triangle cuts the circle in 3 equal arcs, isnt the circumference 24+24+24=72? I tried to solve it this way and failed...

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:
24 = 2/3c
c = 36.

Thus:
Ï€d = 36.
d ≈ 36/π = a little less than 12.

The correct answer is C.

Hi Brent,

It's all much clearer now. I guess I just misunderstood the notation of the arc and didn't realize that arc ABC=arc AB + arc BC.

Thanks a lot,
Elena