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DIRECTIONS for the question: Solve the following question and mark the best possible option.
The incircle of an equilateral triangle is the same as the circumcircle of a square of side 1 unit. What is the area of the triangle in square units?
  • a)
    √3 / 4
  • b)
    3√3 / 2
  • c)
    √3 / 2
  • d)
    3√3
Correct answer is option 'B'. Can you explain this answer?
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DIRECTIONSfor the question:Solve the following question and mark the b...
The diagonal of the square is √2, which is also the diameter of the circumcircle. So the radius of the circumcircle is 1/√2. This is also the radius of the in circle of the equilateral triangle. We know that the inradius of an equilateral triangle is 1/3 the height of the triangle. So, the height of the equilateral triangle is 3/√2. Now, in Triangle ABC, which is a 30°–60°–90°, if AB = 1/√2, then BC = √3/√2. So, the area of the equilateral triangle is 3/√2 × √3/√2 = 3√3/2 square units.
Alternately,
area of  triangle ABC = ½ × 1/√2 × √3/√2 = √3/4. So the area of the equilateral triangle is 6 × √3/4 = 3√3/2.
Students can also choose to find the side of the equilateral triangle and then find its area as √3s2/4.
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Most Upvoted Answer
DIRECTIONSfor the question:Solve the following question and mark the b...
(a) $\frac{4\sqrt{3}}{3}$ square units
(b) $\frac{3}{2}$ square units
(c) $\frac{2\sqrt{3}}{3}$ square units
(d) $\frac{\sqrt{3}}{2}$ square units

To solve this problem, we can start by finding the radius of the incircle of the equilateral triangle. Since the incircle is the same as the circumcircle of the square, we know that the radius of the incircle is equal to the radius of the circumcircle of the square.

The radius of the circumcircle of the square is equal to half the length of its diagonal. Since the square has side length 1 unit, its diagonal is $\sqrt{2}$ units long. Therefore, the radius of the incircle is $\frac{\sqrt{2}}{2}$ units.

Next, we can find the height of the equilateral triangle by drawing an altitude from one of the vertices to the base. This altitude splits the equilateral triangle into two congruent 30-60-90 triangles. The hypotenuse of each of these triangles is equal to the side length of the equilateral triangle, so it is 1 unit long. The height of the equilateral triangle is then equal to the length of the short leg of the 30-60-90 triangle, which is $\frac{\sqrt{3}}{2}$ units.

Finally, we can calculate the area of the equilateral triangle using the formula $A = \frac{1}{2}bh$, where $b$ is the base (which is equal to the side length of the equilateral triangle) and $h$ is the height. Plugging in the values we found, we get $A = \frac{1}{2} \cdot 1 \cdot \frac{\sqrt{3}}{2} = \boxed{\frac{\sqrt{3}}{4}}$ square units.

Therefore, the correct option is (d) $\frac{\sqrt{3}}{2}$ square units.
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In a certain training institute which had exactly 200 students, each student was given exactly one question from one among the following categories: Reasoning, Mathematics, Grammar, Interpretation and Comprehension.Chart 1 gives the distribution of questions among the students.However, since the students were not happy with the type of questions each one had received, they exchanged their questions with other students in such a way that none of them had a question of the same category as they had earlier.Chart 2 shows the distribution of number of students who initially had questions from Reasoning category and exchanged their questions across students of various categories.Further, exactly 8 students who originally had questions on Interpretation now have questions on Reasoning with them. Also 8 students who initially had questions on comprehension now have questions on Interpretation category.Also out of the students who initially had questions on Grammar, 3 students now have questions on Reasoning and 16 have questions on Mathematics category.If out of the students who initially had questions on Mathematics with them, the number of students who now have questions on Comprehension is the maximum possible; then out of the students who now have questions on Grammar category what is the maximum possible number of students who initially had questions on Comprehension category? Correct answer is '16'. Can you explain this answer?

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