Solve the following question and mark the best possible option.A square, S1, circumscribes the circumcircle of an equilateral triangle of side 10 cm. A square, S2, is inscribed in the incircle of the triangle. What is the ratio of the area of S1 to the area of S2?a)4 : 1b)32 : 1c)8 : 1d)2 : 1Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Option 3

The height of the equilateral triangle is 5√3 cm.
Since the height is also the median, we know that the circum-radius is 2/3 × 5√3 = 10√3/3 and the in-radius is 1/3 × 5√3 = 5√3/3.
The diameter of the circumcircle is the side of square S1.

So the area of S1 is (2 × 10√3/3)² = 1200/9.
The diameter of the in-circle is the diagonal of square S2.

So the area of S2 is ½ × (2 × 5√3/3)2 = 300/18.
Thus the ratio of areas S1 : S2 is 1200/9 : 300/18 = 8 : 1.

Answer

Option 3

The height of the equilateral triangle is 5√3 cm.
Since the height is also the median, we know that the circum-radius is 2/3 × 5√3 = 10√3/3 and the in-radius is 1/3 × 5√3 = 5√3/3.
The diameter of the circumcircle is the side of square S1.

So the area of S1 is (2 × 10√3/3)² = 1200/9.
The diameter of the in-circle is the diagonal of square S2.

So the area of S2 is ½ × (2 × 5√3/3)2 = 300/18.
Thus the ratio of areas S1 : S2 is 1200/9 : 300/18 = 8 : 1.

Answer

To solve this question, we need to understand the properties of the circumcircle and incircle of an equilateral triangle.

Properties of the Circumcircle:
- The circumcircle of an equilateral triangle passes through all three vertices of the triangle.
- The center of the circumcircle is equidistant from all three vertices of the triangle.
- The radius of the circumcircle is equal to the distance between the center and any vertex of the triangle.

Properties of the Incircle:
- The incircle of an equilateral triangle is tangent to all three sides of the triangle.
- The center of the incircle is equidistant from all three sides of the triangle.
- The radius of the incircle is equal to the distance between the center and any side of the triangle.

Let's solve the problem step by step:

1. Finding the radius of the circumcircle:
Since the circumcircle passes through all three vertices of the equilateral triangle, the radius of the circumcircle is equal to the distance between the center and any vertex of the triangle. Let's assume this distance as 'r'.
In an equilateral triangle, the distance between the center and any vertex is equal to the height of the triangle. Using the formula for the height of an equilateral triangle, which is given by h = (√3/2) * side, we can find the radius of the circumcircle.

Given side of the equilateral triangle = 10 cm
Height of the equilateral triangle = (√3/2) * 10 = 5√3 cm

Therefore, the radius of the circumcircle = r = 5√3 cm.

2. Finding the side of the square S1:
Since square S1 circumscribes the circumcircle of the equilateral triangle, the diagonal of square S1 is equal to the diameter of the circumcircle, which is twice the radius.
Diagonal of square S1 = 2 * r = 2 * 5√3 = 10√3 cm

Therefore, the side of square S1 = (10√3)/√2 = 5√6 cm (dividing the diagonal by √2 to get the side length)

3. Finding the side of the square S2:
Since square S2 is inscribed in the incircle of the equilateral triangle, the diagonal of square S2 is equal to the diameter of the incircle, which is twice the radius.
Diagonal of square S2 = 2 * r = 2 * 5√3 = 10√3 cm

Therefore, the side of square S2 = (10√3)/√2 = 5√6 cm (dividing the diagonal by √2 to get the side length)

4. Finding the ratio of the areas of S1 and S2:
Area of square S1 = (side of S1)^2 = (5√6)^2 = 150 cm^2
Area of square S2 = (side of S2)^2 = (5√6)^2 = 150 cm^2

Therefore, the ratio of the area of S1 to the area of S2 = 150/150 = 1/1 = 1 : 1.

Hence, the correct answer is option 'D' - 2 : 1.

Similar CAT Doubts

DIRECTIONSfor 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 circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?

A regular hexagon (H1) of side 8 cm is inscribed in a circle(C1). Another circle(C2) is inscribed in the regular hexagon (H1) and it circumscribes another regular hexagon(H2) and so on. Find the sum of the areas of all such possible regular hexagons.

An isosceles right-angled triangle PQR is drawn inside an equilateral triangle ABC such that the circumcircle of PQR is the incircle of ABC. If the area of the triangle PQR is 27 cm2cm, what is the length of side CA (in cm)? Correct answer is '18'. Can you explain this answer?

A circle is inscribed in a square of side 10cm. Another circle circumscribes the square. What is the ratio of the areas enclosed between the outer circle and square to the are enclosed between the inner circle and square?

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