Geometry in Math Competition —Shuangcheng

This week I switched back to problem solving practices on Harvard-MIT questions, and I would like to introduce another category of problems that I have been solving: Geometry. Geometry problems frequently show up in different kinds of mathematics competitions, including the Harvard-MIT competitions. According to the archives of the General tests from recently years, three out of ten questions are geometric problems, which is a high ratio. Therefore, it is important to prepare for geometric problems in order to succeed in the exam. This week I have worked on several geometric problems from Harvard-MIT competition. Below are a couple of questions that I chose to show my solutions.

3. [3] ABCD is a rectangle with AB = 20 and BC = 3. A circle with radius 5, centered at the midpoint of DC, meets the rectangle at four points: W, X, Y , and Z. Find the area of quadrilateral WXYZ.

This question appears as the third question on the general test  in November 2012. Personally I think that this question is easy and pretty straightforward. Therefore, I did not write out all the detailed procedures in my picture below.

Geometry 2

6. [5] ABCD is a parallelogram satisfying AB = 7, BC = 2, and ∠DAB = 120◦
. Parallelogram ECFA is contained in ABCD and is similar to it. Find the ratio of the area of ECF A to the area of ABCD.

Geometry 1

 This question is also from the general test in 2012. Apparently it is more difficult to solve this one than the last one. Therefore, I wrote a more thorough solution for this question. What I would like to emphasize in this questions is that the ratio of the areas of similar figures is equal to the square of the ratio of their side. 

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