Difference between revisions of "2018 AMC 8 Problems/Problem 22"
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Point is the midpoint of side in square and meets diagonal at The area of quadrilateral is What is the area of
Let the area of be . Thus, the area of triangle is and the area of the square is .
By AA similarity, with a 1:2 ratio, so the area of triangle is . Now consider trapezoid . Its area is , which is three-fourths the area of the square. We set up an equation in :
Solving, we get . The area of square is .
We can use analytic geometry for this problem.
Let us start by giving the coordinate , the coordinate , and so forth. and can be represented by the equations and , respectively. Solving for their intersection gives point coordinates .
Now, ’s area is simply or . This means that pentagon ’s area is of the entire square, and it follows that quadrilateral ’s area is of the square.
The area of the square is then .
Note that triangle has half the area of the square and triangle has th. Thus the area of the quadrilateral is th the area of the square. The area of the square is then .
Extend and to meet at . Drop an altitude from to and call it . Also, call . As stated before, we have , so the ratio of their heights is in a ratio, making the altitude from to . Note that this means that the side of the square is . In addition, by AA Similarity in a ratio. This means that the side length of the square is , making .
Now, note that . We have and Subtracting makes We are given that so Therefore, so our answer is - moony_eyed
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