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co-ordinate geometry

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co-ordinate geometry

by ritind » Tue Dec 04, 2012 11:28 pm
Point a is the center of both a circle and a square. The circle, which is fully shown above, is inscribed in the square and the circle is tangent on all sides with the square, which is only partially shown and has both the x-axis and the y-axis as sides. The origin (0,0) is the bottom-left corner of the square and the line DE is a diagonal of the square. If the x-coordinate of point a is x1, what is the area of the gray shaded region between the circle and the origin (0,0)?
A) .25(x1)2[4 - π]
B) x12 - x12Ï€
C) .25[2(x1)2 - x12Ï€]
D) 4(x1)2 - x12Ï€
E) x12 - x12.5Ï€

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by tsalagi » Tue Dec 04, 2012 11:43 pm
It is often helpful to draw auxiliary lines in a geometry problem. In particular, you should *always* draw any radius that will intersect a tangent to a circle. In this problem, let's draw radii to the tangent points that correspond roughly to where F and C are labeled in your diagram. The area we are looking for is that of a shape for which you won't necessarily have memorized a formula. That's okay though because we can reinterpret it as a square with a quarter circle cut out of it, both of which you probably know how to find the area of. The square has side length x1, which gives it an area of x1^2. The quarter circle has radius x1, which gives it an area of pi*x1^2 / 4 (using the formula for area of a circle but remembering to divide by 4 because we only have a quarter circle). Now the area we are looking for is x1^2 - pi*x1^2 / 4, and we just need to find an answer choice that is equivalent. If I'm interpreting your typography correctly, it appears that A does just what we're looking for.
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by ritind » Wed Dec 05, 2012 12:35 am
A is the correct choice
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