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What is the area of the shaded figure?

admin2011-01-18  101
问题 What is the area of the shaded figure?
选项 A、 
B、 
C、 
D、 
E、 
答案A
解析 This figure may look intimidating, but it’s actually quite harmless. To tackle it, you must first recognize that it can be split into two smaller triangles, as shown in the diagram. Once you’ve separated the triangles, you can start labeling the lengths. The lower triangle has two 45° angles, so it is clearly a right isosceles triangle. That means that we should be able to find the area quickly. An intermediate step will be calculating the lengths of the (equal) sides, which we’ll designate as x. That will also have a follow-on purpose. We’ll be able to use the length x with the full length of the left side of the figure to calculate the length of the rising side of the smaller triangle. First, though, we’ll get back to the lower triangle and the Pythagorean Theorem:

   Now we can use the result for x to find that smaller length:

   The triangle may not look familiar yet, but the angle at the right is a giveaway. Since it’s a 120° angle, the supplementary angle θ must be 60°. This is because interior and exterior angles along the same line segment must always sum to 180°. Since another angle in the triangle is right (90°), we now can assert that angle σ measures 30°, since all the angles inside a triangle must sum to 180° as well. This means that we have a 30°-60°-90° triangle. One needs to be careful at this point, as it is easy to misidentify the lengths in one of these. Remember, it is not that the lengths are always 1, , and 2; but that they are always in the ratio of 1::2. If the long nonhypotenuse side has a length of one-half of , then the other sides should be halved as well. The short side will have length 1/2, and the hypotenuse will have length 1.

   At this point, as shown in the last diagram, we can ignore all our calculations but those for a few lengths. The area of the original figure is found with just the four numbers in black:

   or answer choice A. That’s about it. There are other ways of finding the area, but if you can handle the steps here, you should have little trouble with GMAT geometry.
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