Bunuel wrote:
In the figure above, ABCD is a rectangle inscribed in a circle. If the length of AB is three times the length of AD, then what is the ratio of the area of the rectangle to the area of the circle? (Figure not drawn to scale.)
A. 1:2
B. 3:2π
C. 2:5
D. 4:3π
E. 6:5π
Kudos for a correct solution.Attachment:
rectangle-circle.gif
Let's say AD has length 1 and AB has length 3 (this also means that side CD also has length 3)
Since the blue triangle is a right triangle, we can use the Pythagorean Theorem to find the length of the 3rd side (the hypotenuse)
If we let x = the length of the hypotenuse, we can write: 1² + 3² = x²
Simplify: 1 + 9 = x²
So, x² = 10, which means x = √10
At this point, we can see that the diameter of the circle is √10, which means the radius is
(√10)/2What is the ratio of the area of the rectangle to the area of the circle?Area of rectangle = (length)(width)
= (
1)(
3)
= 3
Area of circle = π(radius)²
= π(
√10)/2)²
= π(10/4)
= π(5/2)
= 5π/2
So, the RATIO = 3/(5π/2)
= (3)(2/5π)
= 6/5π
= 6 : 5π
Answer: E
Cheers,
Brent
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