The Figure below Shows an Equilateral Triangle ABC Tangent GMAT Problem Solving

Question: The figure below shows an equilateral triangle ABC tangent to the circle O at three points. If the perimeter of triangle ABC is 18, the area of the circle =

1. \(2\pi\)
2. \(\frac{5\pi}{2}\)
3. \(2\sqrt2\pi\)
4. \(3\pi\)
5. \(2\pi\sqrt3\)

Correct Answer: (D)

Solution and Explanation:
Approach Solution 1:
The in-radius of an equilateral triangle is √3* (side)/6
(In-radius is 1/3 the length of an altitude, because each altitude is also a median of the triangle)

Perimeter of the given triangle = 18 units

Each side of the triangle = 18/3 = 6 units

In radius = \(\frac{\sqrt3}{6}*6=\sqrt3\)

Hence area = \(\pi r^2=\pi*\sqrt3*\sqrt3=3\pi\)

Approach Solution 2:

Area = in-radius * semi-perimeter

So in-radius =\(\sqrt3\)

Area =\(3\pi\)

Approach Solution 3:

To find the area of the circle, we need to find its radius.

Let’s draw a radius from any of the three points of tangency, and then create a right triangle.

Since ,ㄥBAC = 60° then ㄥOAD = 30° and Δ AOD is a 1 : √3 : 2 triangle. The question states that the perimeter of Δ ABC is 18, AD = 3.

Let OD = x

Therefore, 3/x = √3/1
=> x√3= 3
=> x = 3/√3 or √3. The circle's radius .

Therefore, the radius of the circle = √3
Hence, area of circle = π(√3)^2 = 3π

“The figure below shows an equilateral triangle ABC tangent to the circle O at three points. If the perimeter of triangle ABC is 18, the area of the circle =”- 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. GMAT Quant practice papers include various types of questions that allow the candidates to improve their mathematical learning.

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