AMC12 2019 A

AMC12 2019 A · Q5

AMC12 2019 A · Q5. It mainly tests Area & perimeter, Coordinate geometry.

Two lines with slopes $\frac{1}{2}$ and 2 intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x + y = 10$?

两条斜率分别为 $\frac{1}{2}$ 和 2 的直线相交于点 $(2,2)$。这两条直线与直线 $x + y = 10$ 围成的三角形的面积是多少？

(A) 4 4

(B) $4\sqrt{2}$ 4$\sqrt{2}$

(D) 8 8

(E) $6\sqrt{2}$ 6$\sqrt{2}$

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Let $P(2,2)$ be the intersection point. The two lines have equations $y=\frac{1}{2}x+1$ and $y=2x-2$. They intersect $x+y=10$ at $A(6,4)$ and $B(4,6)$. Consider $\overline{AB}$ to be the base of the triangle; then the altitude of the triangle is the segment joining $(2,2)$ and $(5,5)$. By the Distance Formula, the area of $\triangle PAB$ is $\frac{1}{2}\cdot\sqrt{(6-4)^2+(4-6)^2}\cdot\sqrt{(5-2)^2+(5-2)^2}=\frac{1}{2}\cdot2\sqrt{2}\cdot3\sqrt{2}=6.$ Note: The area of the triangle with vertices $(2,2)$, $(6,4)$, and $(4,6)$ can be calculated in a number of other ways, such as by enclosing it in a $4\times4$ square with sides parallel to the coordinate axes and subtracting the areas of three right triangles; by splitting it into two triangles with the line $y=4$; by the shoelace formula: $\frac{1}{2}\cdot\left|(2\cdot4+6\cdot6+4\cdot2)-(6\cdot2+4\cdot4+2\cdot6)\right|=6;$ or by observing that there are $4$ lattice points in the interior of the triangle and $6$ lattice points on the boundary, and using Pick’s Formula: $4+\frac{6}{2}-1=6.$

答案（C）：设 $P(2,2)$ 为交点。两条直线的方程为 $y=\frac{1}{2}x+1$ 和 $y=2x-2$。它们与 $x+y=10$ 的交点分别为 $A(6,4)$ 和 $B(4,6)$。取 $\overline{AB}$ 为三角形的底，则三角形的高为连接 $(2,2)$ 与 $(5,5)$ 的线段。由距离公式，$\triangle PAB$ 的面积为 $\frac{1}{2}\cdot\sqrt{(6-4)^2+(4-6)^2}\cdot\sqrt{(5-2)^2+(5-2)^2}=\frac{1}{2}\cdot2\sqrt{2}\cdot3\sqrt{2}=6.$ 注：顶点为 $(2,2)$、$(6,4)$、$(4,6)$ 的三角形面积还可用多种方法计算，例如：将其包含在边与坐标轴平行的 $4\times4$ 正方形中并减去三个直角三角形的面积；用直线 $y=4$ 将其分成两个三角形；使用鞋带公式： $\frac{1}{2}\cdot\left|(2\cdot4+6\cdot6+4\cdot2)-(6\cdot2+4\cdot4+2\cdot6)\right|=6;$ 或注意到三角形内部有 $4$ 个格点、边界上有 $6$ 个格点，并用皮克公式： $4+\frac{6}{2}-1=6.$

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