In the figure above, an equilateral triangle is inscribed

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 am; Followed by:3 members

In the figure above, an equilateral triangle is inscribed

by VJesus12 » Sat Jan 05, 2019 7:03 am

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In the figure above, an equilateral triangle is inscribed in a circle. If the arc bounded by adjacent corners of the triangle is between 4Ï€ and 6Ï€ long, which of the following could be the diameter of the circle?

(A) 6.5
(B) 9
(C) 11.9
(D) 15
(E) 23.5

[spoiler]OA=D[/spoiler]

Source: Manhattan GMAT

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Source: Beat The GMAT — Problem Solving | Sat Jan 05, 2019 7:03 am

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[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pm; Location: Palo Alto, CA; Thanked: 2867 times; Followed by:511 members; GMAT Score:800

by [email protected] » Sun Jan 06, 2019 11:33 am

Hi VJesus12,

We're told that in the figure above, an EQUILATERAL triangle is inscribed in a circle and the arc bounded by adjacent corners of the triangle is between 4Ï€ and 6Ï€ long. We're asked which of the following COULD be the diameter of the circle. As scary as this question might look, it's based on a couple of standard Geometry rules, so you can answer it with just a little work.

To start, an equilateral triangle has 3 equal angles - and since the triangle is inscribed in the circle, each of the three 'arc pieces' is equal in length. Thus, the total of those three arcs (re: the circumference of the circle) is between (3)(4Ï€) and (3)(6Ï€). If the total circumference is between 12Ï€ and 18Ï€, then we can 'work backwards' to find the possible diameter....

12Ï€ = 2Ï€(R) = Ï€D..... Diameter = 12
18Ï€ = 2Ï€(R) = Ï€D..... Diameter = 18

There's only one answer that's between 12 and 18...

Final Answer: D

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8086; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

by Scott@TargetTestPrep » Fri Feb 01, 2019 5:23 pm

VJesus12 wrote:

In the figure above, an equilateral triangle is inscribed in a circle. If the arc bounded by adjacent corners of the triangle is between 4Ï€ and 6Ï€ long, which of the following could be the diameter of the circle?

(A) 6.5
(B) 9
(C) 11.9
(D) 15
(E) 23.5

[spoiler]OA=D[/spoiler]

Source: Manhattan GMAT

Since each arc is bounded by adjacent corners of the triangle representing 1/3 of the circumference, the range of values of the circumference is:

Minimum:

1/3(C) = 4Ï€

C = 12Ï€, so the diameter would be 12.

Maximum:

1/3(C) = 6Ï€

C = 18Ï€, so the diameter would be 18.

The diameter is between 12 and 18, so a possible diameter is 15.

Answer: D

Scott Woodbury-Stewart
Founder and CEO
[email protected]