Problem Solving

Can anyone help with this question

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

recko2430 wrote:Can anyone help with this question

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

ABC creates three equal arcs on the circle. ( 60-60-60)

arc ABC : 24 so= Perimeter is: 3/2 x 24 = 36

36= 2pir OR Dpi ( since D=2r)

So, D = 36/pi ( pi=3.14...)

So D is slightly smaller than 12:

Hence C

would pls explain this more,

how did u get 3/2*24

My approach was:

It is an equilateral triangle so there are three equal arcs.

The length of arc abc would be 2/3 of the circumference (circumference = pi(d)).

So this means:
24 = 2/3 * pi(d)
24/pi(d)= 2/3
72 = 2pi(d)
36 = pi(d)
36/pi = d

~11 = d

Hi,

Where are you getting the 2/3 from? Would appreciate if you may please explain the theory behind it. Thank you!!

Take a look at this diagram:
https://etc.usf.edu/clipart/49900/49948/ ... ion_md.gif

Because it's an equilateral triangle, the lengths of arcs AB, BC, and AC are all equal. That means each is 1/3 of the total circumference of the circle.

arc ABC is formed by joining arcs AB and BC, which each represent 1/3 of the circumference. Therefore, arc ABC is 2/3 of the circumference.

recko2430 wrote:Can anyone help with this question

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, �d = 36.
d ≈ 11.

The correct answer is C.