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
(E) Let AD = x. AB = 3x. Area of the rectangle is (AD) × (AB) = x × (3x) = 3x².
Using the Pythagorean theorem, take AB and AD to get the diameter AC of the circle. x² + (3x)² = AC² x² + 9x² = AC² 10x² = AC² AC = x√10 and the radius is [x√10] / 2.
The area of a circle is πr², so the area is π [(x√10) / 2]² = (10πx²)/4 = (5πx²)/2.
The ratio is: 3x² : (5πx²)/2 = 3 : 5π/2 = 6 : 5π
The correct answer is E. ---------- I was wondering why you couldn't assume that point O was the center of the circle. Does the question have to directly mention that fact, because I assumed that it was the center and therefore took OD and OA as being radii. Which ultimately gives you an answer of 3:π.
I was wondering why you couldn't assume that point O was the center of the circle. Does the question have to directly mention that fact …
No, you cannot. Anything that is NOT stated by the question statement or specifically marked on graphics is considered unknown. (However basic facts, such as: lines, which look straight, are straight; how the points are notated, etc. can be assumed based on the graphics.)
But maybe you could prove (based on the given facts) that O is the center of the circle? Unfortunately no. There are no grounds for that.
… , because I assumed that it was the center and therefore took OD and OA as being radii. Which ultimately gives you an answer of 3:π.
Even if O was the center of the circle, then 3:π would still NOT be the correct answer. OA and OD would be the radii indeed, but the triangle AOD would be just an isosceles one. And as the explanation shows, the radius differs from the length of AD, so AOD would NOT be an equilateral triangle.
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