Problem Solving

Looking at the arc, I see that it covers two sides of the triangle.

Given that the triangle is equilateral, it seems clear that the length of the arc is 2/3 of the circumference of the circle.

So if 24 is 2/3 of the circumference, then 1/3 is 12 and the entire circumference, C, is 3 x 12 = 36.

C = Pi x the diameter, D. C = PiD

Pi is just over 3. 36/3 = 12 36/3.14 is a little less, approximately 11.

Choose C.

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.

Two-thirds of the circumference = 24
So one third is 12
So the whole circumference, C = 36
Diameter, D = C/pi
pi = 3 (approx)
so D = 36/3 = 12 approx.

ANSWER C