AMC 8 2018 Problem 22
AMC8 2018 年第 22 题
四边形与多边形★★★☆☆
Point is the midpoint of side in square , and meets diagonal at . The area of quadrilateral is 45. What is the area of ?
在正方形 中， 是边 的中点， 和对角线 交于点 。四边形 的面积是 45，问 的面积是多少？
- A.100
- B.108
- C.120
- D.135
- E.144
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
We can use analytic geometry for this problem. Let us start by giving the coordinate (0,0), the coordinate (0,1), and so forth. and can be represented by the equations y=-x+1 and y=2x-1, respectively. Solving for their intersection gives point coordinates . Now, we can see that '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 .
Let be the sides of the square. has half the area of the square. has base equal to half the square side length, and by AA Similarity with , it has the height, so has th of the area of square( * * according to the triangle area formula). Thus, the area of the quadrilateral is of the area of the square. We get because since the side length ratio of to is 2, the area ratio would be 4. Hence, . The area of is . Thus, the area of equals to the area of minus the area of , which is of (the area of the square). We can write the equation: Solving the equation, we get The area of the square is then . edit by abirgh
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 square's side length is , making 3h=2x. Now, note that [ADEF]=[XAB]-[XDE]-[ABF]. We have and Subtracting makes We are given that [ADEF]=45, so Therefore, so our answer is
~ moony_eyed ~ Minor Edits by WrenMath
Solution with Cartesian and Barycentric Coordinates: We start with the following: Claim: Given a square , let be the midpoint of and let . Then . Proof: We use Cartesian coordinates. Let be the origin, A=(0,1),C=(0,1),B=(1,1). We have that and are governed by the equations y=-x+1 and y=2x-1, respectively. Solving, . The result follows. Now, we apply Barycentric Coordinates w.r.t. . We let A=(1,0,0),D=(0,1,0),C=(0,0,1). Then . In the barycentric coordinate system, the area formula is $[XYZ]=
$ where $$ is a random triangle and $$ is the reference triangle. Using this, we find that
Let [FEC]=x so that [ACD]=45+x. Then, we have , so the answer is 2(45+9)=108.
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