AMC 8 2019 Problem 21
AMC8 2019 年第 21 题
坐标几何★★★☆☆
What is the area of the triangle formed by the lines y=5 , y=1+x , and y=1-x ?
由三条直线 y=5，y=1+x 和 y=1-x 所包围的三角形的面积是多少？
- A.4
- B.8
- C.10
- D.12
- E.16
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
First, we need to find the coordinates where the graphs intersect. We want the points x and y to be the same. Thus, we set 5=x+1, and get x=4. Plugging this into the equation, y=1-x, y=5, and y=1+x intersect at (4,5), we call this line x. Doing the same thing, we get x=-4. Thus, y=5. Also, y=5 and y=1-x intersect at (-4,5), and we call this line y. It's apparent the only solution to 1-x=1+x is 0. Thus, y=1. y=1-x and y=1+x intersect at (0,1), we call this line z. Using the Shoelace Theorem we get:
So, our answer is 16. We might also see that the lines and are mirror images of each other. This is because, when rewritten, their slopes can be multiplied by -1 to get the other. As the base is horizontal, this is an isosceles triangle with base 8, as the intersection points have a distance of 8. The height is 5-1=4, so by WrenMath Warning: Do not use the distance formula for the base then use Heron's formula. It will take you half of the time you have left!
Graphing the lines, using the intersection points we found in, we can see that the height of the triangle is 4, and the base is 8. Using the formula for the area of a triangle, we get , which is equal to 16.
y = x + 1 and y = -x + 1 have -intercepts at (0, 1) and slopes of 1 and -1, respectively. Since the product of these slopes is -1, the two lines are perpendicular. From y = 5, we see that (-4, 5) and (4, 5) are the other two intersection points, and they are 8 units apart. By symmetry, this triangle is a 45-45-90 triangle, so the legs are each and the area is .
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