Problem Solving

Mjkourtis wrote:In the figure, equilateral triangle ABC is inscribed in a circle. If the length of arc ABC is 24, what is the diameter?
A. 5
B. 8
C. 11
D. 15
E. 19

Hi Mjkourtis!

Fun geometry problem, but I am going to assume that the question SHOULD say one of the following:

(1) the length of the arc is actually 24(Pi) (but then no answer choices work)
(2) the answers should ALL be divided by Pi (e.g. 5/Pi, 8/Pi) - but again, this problem doesn't work
(3) the answer is the APPROXIMATE diameter - in this case the answer could be one of the choices, but it isn't a good approximation.

So please check the question so we can make sure we're right!

Now, let's pretend that everything is AS WRITTEN and just go with the approximation...

Anytime we are told about arc lengths, we immediately want to think about what PROPORTION of the circle the arc covers. Notice that they call Triangle ABC and equilateral triangle - that means that is has 3 equal sides, and therefore the distances around the circle from vertex to vertex should also be equal (we can prove this with the relationship between central and inscribed angles so I'll throw that in at the bottom for extra).

So, if the 3 arcs AB, BC, and CA are the same, then each one is 1/3 of the total distance around, or 1/3 of the total circumference. Well, AC is the sum of 2 of those, so that would mean that AC is 2/3 of the entire circumference. SO lets catalog what we know:

Circumference is C=Pi*D where D is the diameter.
(2/3)*(C) = 24

So (2/3)*(Pi*D) = 24
Pi*D = 24*(3/2)
Pi*D = 36
D = 36/Pi

If we are going to approximate this, we would divide 36 by something a bit larger than 3 (3.14) so this would be something smaller than 12 (actually 11.46) so the only answer that is a little less than 12 would have to be C.

Hope this helps!

Whit

Mjkourtis wrote:In the figure, equilateral triangle ABC is inscribed in a circle. If the length of arc ABC is 24, what is the diameter?
A. 5
B. 8
C. 11
D. 15
E. 19

As triangle ABC is equilateral, arc ABC is 2/3 of the circumference of the circle.
Hence, circumference of the circle = 24*3/2 = 36

Hence, diameter of the circle = 36/π = Slightly less than 36/3 ≈ 11

The correct answer is C.