Question: A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

1. 20
2. 45
3. 60
4. 55
5. 77

Answer:
Solution and Explanation:
Approach Solution 1:
To solve this GMAT problem-solving question, you must use the information given in the question. The problems in this group come from many different areas of mathematics. This one has a lot to do with circles. The options are set up in a way that makes it hard to pick the best one. The candidates must know the right way to get the response they need. Only one of the five choices given is correct.
Given in the question that in order for the two figures to touch exactly three times, one on each side of the triangle, a circle must be inscribed within an equilateral triangle. Which of the following most closely approximates the proportion of the triangle's area that is contained within the circle?
the Radius of a circle inscribed = \(\sqrt3a/6\)
The altitude of the equilateral triangle is known to be \(\sqrt3a/2\)
In addition to being the median, the altitude will pass through the center of the circle (since it is an equilateral triangle). We know that the ratio of the centroid to the median is 2:1. Due to symmetry, the centroid will be the center of the circle, as each median will pass through it. Therefore, the circle's radius will equal one-third of its altitude.
Radius = \(\sqrt3a/2 * 1/3 = \sqrt3a/6\)
Let the side of the equilateral triangle = a
Area of triangle = \(\sqrt3a^2/4\)
Radius of inscribed circle = \(a\sqrt3/6\)
Area of inscribed circle = \({\pi}a^2/12\)
Percentage area = \({\pi}a^2/12/(\sqrt3a^2/4) = 0.628\)
The correct choice is C.

Approach Solution 2:
To solve this GMAT problem-solving question, you must use the information given in the question. The problems in this group come from many different areas of mathematics. This one has a lot to do with circles.
The options are set up in a way that makes it hard to pick the best one. The candidates must know the right way to get the response they need. Only one of the five choices given is correct.
.
Given in the question that in order for the two figures to touch exactly three times, one on each side of the triangle, a circle must be inscribed within an equilateral triangle. Which of the following most closely approximates the proportion of the triangle's area that is contained within the circle?
so now area of an equilateral triangle = √3/4∗a^2 —---1
and area of the triangle is also equal to Area of triangle AOC+Area of AOB + Area of BOC = 1/2∗a∗r ( r is the height of the individual triangle) ---------2
from equation 1 & 2 above
√3/4 * a^2 = 3∗ ½ ∗ a ∗ r
from here we can get the value of an i.e. a=2√3 ∗ r ---------3
Now, In the question, we need to find out the area of a circle containing the area of the equilateral triangle.
which is equal to π∗r^2 / (3∗1/2∗(2√3 * r^2)
------substituting the value of a from equation 3
=π/(3∗√3)
≈3.14/ (3∗1.72)
≈2/3 ≈ 0.66
The correct choice is C.

Approach Solution 3:
You must use the data provided in the question to resolve this GMAT problem-solving issue. These issues span a wide range of mathematical disciplines. This one involves math a lot.
It's challenging to choose the best option because of how the options are presented. The candidates must understand the proper procedure to obtain the necessary response. Out of the five options provided, only one is
accurate.
Given that in order for the two figures to touch exactly three times, one on each side of the triangle, a circle must be inscribed within an equilateral triangle. Which of the following most closely approximates the proportion of the triangle's area that is contained within the circle?
If the question requires percentages or ratios, we can always try a number, such as "6" for the side of the equilateral triangle.
We must determine the triangle's area and the circumscribed circle's area.
Consequently, (Area of the circle inscribed/Area of the triangle)*100
The circle's area is (pi)*r2
the triangle's area equals 1/2 * base * height
Refer to the illustration,
Therefore, the triangle's area is 12 6 3 (root 3) = 9 root 3,
Remember also that the radius of a circle is always one-third the height of an equilateral triangle (This is because, Medians of the triangle intersects at 1:2 ratio).
Therefore, the radius is 1/3(3 root 3) = root 3
Therefore, the area of a circle is pi * (root 3)2 = 3 * pi.
We can approximate the value of "pi" as "3" because the answer options are sufficiently diverse. Consequently, the area of the circle is roughly 9.
Now, we can approximate the triangle's area as 9 root 3 * 1.7 = 15.3 We can therefore further approximate it as 15
(Area of the circle inscribed/Area of the triangle)*100 (9/15)*100 yields the percentage.
This is sixty percent.
Therefore, the correct answer is C.

“A circle is inscribed in an equilateral triangle, Find area" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.
To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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