Let x be the area of a square inscribed in a circle of radius r and y be the area of an equilateral triangle inscribed in the same circle. Which one of the following is correct?
- A. 9x^{2}=16y^{2}
- B. 27x^{2}=64y^{2} ✓
- C. 36x^{2}=49y^{2}
- D. 16x^{2}=21y^{2}
Correct Answer: B. 27x^{2}=64y^{2}
Explanation
For the inscribed square, the diagonal is 2r. Side a = r\sqrt{2}, so area x = a^2 = 2r^2. For the inscribed equilateral triangle, the circumradius r = \frac{s}{\sqrt{3}}, so side s = r\sqrt{3}. Area y = \frac{\sqrt{3}}{4}s^2 = \frac{3\sqrt{3}}{4}r^2. The ratio \frac{x}{y} = \frac{2r^2}{\frac{3\sqrt{3}}{4}r^2} = \frac{8}{3\sqrt{3}}. Squaring both sides: \frac{x^2}{y^2} = \frac{64}{27}, which gives 27x^2 = 64y^2.
Related questions on Mensuration
- A pendulum swings through an angle of 30° and its end describes an arc of length 55 cm. What is the length of the pendulum? (Take $\pi = 22/...
- A conical tent has an angle of 60° at the vertex. If the curved surface area is 100 m^2, then what is the volume of the tent?
- A right circular cone and a hemisphere have equal base and equal volume. What is the ratio of the height of the cone to the radius of the he...
- A wire is in the form of an equilateral triangle with an area of 36\sqrt{3} cm^2. If it is changed into a semicircle, then what is its r...
- Let the area of the largest possible square inscribed in a circle of unit radius be x. Let the area of the largest possible circle inscribed...