What is the Perimeter of an Equilateral Triangle Inscribed GMAT Problem Solving

Question: What is the perimeter of an equilateral triangle inscribed in a circle of radius 4?

1. \(6\sqrt2\)
2. \(6\sqrt3\)
3. \(12\sqrt2\)
4. \(12\sqrt3\)
5. 24

“What is the perimeter of an equilateral triangle inscribed in a circle of radius 4?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer:

Approach Solution (1):

So here’s what diagram looks like:

If we draw lines from the centre of each vertex, we get the following

Since the radii have the length 4, we can add that here:

Now we will draw a line from the centre that is perpendicular to one side of the triangle.

We now have the special 30-60-90 right triangle

Here’s the base version of this special triangle

we can see that the each 30-60-90 trianlge in the diagram is TWICE as big as the base version. So, each side opposite the 60 degree angle must have the length \(2\sqrt3\)

This means one side of the eqiluateral triangle has length \(4\sqrt3\) , so the Perimeter = \(4\sqrt3\) + \(4\sqrt3\)+\(4\sqrt3\) = \(12\sqrt3\)

Correct option: C

Approach Solution (2):

The radius of circum circle of an equilateral triangle = \(\frac{a}{\sqrt3}\)

a is the side of the triangle.

Here: \(\frac{a}{\sqrt3}\) = 4

a = 4 * \( {\sqrt3}\)

Perimeter = 3a = 3 * 4 * \({\sqrt3}\) = 12\({\sqrt3}\)

Correct option: C

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