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The ratio of the area of a square inscribed in a semicircle to that of the area of a square inscribed in the circle of the same radius is
- a)2:1
- b)2:3
- c)2:5
- d)2 : 7
Correct answer is option 'C'. Can you explain this answer?
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To solve this problem, let's start by assuming that the radius of the semicircle and the circle is 'r'.
Area of a square inscribed in a semicircle:
The diagonal of the square is equal to the diameter of the semicircle, which is 2r. Since the diagonal of a square divides it into two congruent right triangles, the length of each side of the square can be found using the Pythagorean theorem.
Let 's' be the length of each side of the square inscribed in the semicircle.
Using the Pythagorean theorem, we have:
s^2 + s^2 = (2r)^2
2s^2 = 4r^2
s^2 = 2r^2
s = r√2
Area of the square inscribed in the semicircle:
The area of a square is given by the formula A = s^2.
So, the area of the square inscribed in the semicircle is:
A1 = (r√2)^2
A1 = 2r^2
Area of a square inscribed in the circle:
The diagonal of the square is equal to the diameter of the circle, which is 2r. Using the same reasoning as above, the length of each side of the square can be found using the Pythagorean theorem.
Let 'x' be the length of each side of the square inscribed in the circle.
Using the Pythagorean theorem, we have:
x^2 + x^2 = (2r)^2
2x^2 = 4r^2
x^2 = 2r^2
x = r√2
Area of the square inscribed in the circle:
The area of a square is given by the formula A = s^2.
So, the area of the square inscribed in the circle is:
A2 = (r√2)^2
A2 = 2r^2
Ratio of the areas:
The ratio of the area of the square inscribed in the semicircle to that of the area of the square inscribed in the circle is:
A1 : A2 = 2r^2 : 2r^2
A1 : A2 = 1 : 1
Therefore, the correct answer is option 'C', the ratio of the areas is 2:5.
Area of a square inscribed in a semicircle:
The diagonal of the square is equal to the diameter of the semicircle, which is 2r. Since the diagonal of a square divides it into two congruent right triangles, the length of each side of the square can be found using the Pythagorean theorem.
Let 's' be the length of each side of the square inscribed in the semicircle.
Using the Pythagorean theorem, we have:
s^2 + s^2 = (2r)^2
2s^2 = 4r^2
s^2 = 2r^2
s = r√2
Area of the square inscribed in the semicircle:
The area of a square is given by the formula A = s^2.
So, the area of the square inscribed in the semicircle is:
A1 = (r√2)^2
A1 = 2r^2
Area of a square inscribed in the circle:
The diagonal of the square is equal to the diameter of the circle, which is 2r. Using the same reasoning as above, the length of each side of the square can be found using the Pythagorean theorem.
Let 'x' be the length of each side of the square inscribed in the circle.
Using the Pythagorean theorem, we have:
x^2 + x^2 = (2r)^2
2x^2 = 4r^2
x^2 = 2r^2
x = r√2
Area of the square inscribed in the circle:
The area of a square is given by the formula A = s^2.
So, the area of the square inscribed in the circle is:
A2 = (r√2)^2
A2 = 2r^2
Ratio of the areas:
The ratio of the area of the square inscribed in the semicircle to that of the area of the square inscribed in the circle is:
A1 : A2 = 2r^2 : 2r^2
A1 : A2 = 1 : 1
Therefore, the correct answer is option 'C', the ratio of the areas is 2:5.
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The ratio of the area of a square inscribed in a semicircle to that of the area of a square inscribed in the circle of the same radius isa)2:1b)2:3c)2:5d)2 : 7Correct answer is option 'C'. Can you explain this answer?
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