A Circle Inscribed in an Equilateral Triangle ABC so that the Point D GMAT Problem Solving

Question: A circle inscribed in an equilateral triangle ABC so that the point D lies on the circle and on the line segment AC and point E lies on the circle and on line segment AB. If line segment AB = 6, what is the area of the figure created by line segments AD, AE and minor arc DE?

1. \(3\sqrt{3}-(\frac{9}{4})\pi\)
2. \(3\sqrt{3} - \pi\)
3. \(6\sqrt{3} - 3\pi\)
4. \(9\sqrt{3} - 3\pi\)
5. Cannot be determined

Correct Answer: C
Solution and Explanation:
Approach Solution 1:

Since it is an equilateral triangle, the required area can be expressed as: [Area of triangle – Area of circle] / 3
Now we know that the side of the triangle = a = 6
Therefore, area of the triangle =\(\frac{\sqrt{3}}{4}a^2 = 9\sqrt{3}\)
Radius of the circle inscribed in an equilateral triangle = \(a\frac{\sqrt{3}}{6} = \sqrt{3}\)

Therefore, area of the circle = \(\pi * r^2 = 3 \pi\)

Thus, required area = \(\frac{9\sqrt{3} - 3 \pi}{3}=3\sqrt{3} - \pi\)

Approach Solution 2:

The circle inscribed in the triangle is a incircle whose radius is given by:-
(Area of triangle (here ABC) / semi-perimeter) = \(\sqrt{3}\) (here).

You are required to calculate the area of green portion:
First of all,
Calculate area of triangle AED(green + yellow) and subtract the area of yellow portion .
area of yellow portion : area of sector DE - area of triangle(DOE)
which is = \(\pi - \frac{3*3\sqrt{3}}{4}\)
Area of triangle AED = \(\frac{9*\sqrt{3}}{4}\)
Now, 
Area of triangle AED (Green + Yellow) - Area of yellow portion = Required area = \(3\sqrt{3} - \pi\)

Correct Option: B

“A circle inscribed in an equilateral triangle ABC so that the point D lies on the circle and on the line segment AC and point E lies on the circle and on line segment AB. If line segment AB = 6, what is the area of the figure created by line segments AD, AE and minor arc DE?”- 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. The candidates can analyse varieties of questions from the GMAT Quant practice papers that will help them to improve their mathematical knowledge.

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