坐标几何

考什么 · What it tests

把几何搬进坐标系用坐标算：两点距离、中点、斜率，以及在网格里求图形面积。会公式就能把「画图量」变成「算数」。

Geometry moved onto a grid and computed with coordinates: distance between points, midpoints, slope, and areas of figures on the plane. The formulas turn "measure the drawing" into "do arithmetic."

需要先会 · Prerequisites

坐标系与点的表示 $(x,y)$ / The coordinate plane and points $(x,y)$
勾股定理（距离公式的底子） / The Pythagorean theorem (behind the distance formula)
平面图形面积 / Areas of plane figures

常见套路 · Common moves

距离就是勾股定理：两点距离 $=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ ，把横差竖差当两条直角边——坐标题最常用的一招。
Distance is just the Pythagorean theorem: the distance between two points is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ , treating the horizontal and vertical gaps as the legs — the most-used move here.
中点取平均：中点坐标 $\displaystyle =\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ ，横纵各取平均。
A midpoint is the average: its coordinates are $\displaystyle \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ , averaging the $x$ 's and the $y$ 's.
求斜三角形面积用「外接矩形减角」：画一个刚好框住它的矩形，减掉四周的直角三角形，避免找底和高。
Find a tilted triangle's area by "bounding box minus corners": draw the smallest rectangle around it and subtract the corner right triangles, avoiding the search for a base and height.

易错点 · Common pitfalls

距离公式里横竖差没平方或漏开方：先平方求和，最后别忘开方。
Skipping the square or the final root in the distance formula: square and add first, then remember to take the root.
斜率把 $\Delta x$ 和 $\Delta y$ 放反：斜率 =\frac{\text{竖直变化}}{\text{水平变化}}=\frac{\Delta y}{\Delta x}，上下别倒。
Flipping $\Delta x$ and $\Delta y$ in the slope: slope $\displaystyle =\frac{\text{vertical change}}{\text{horizontal change}}=\frac{\Delta y}{\Delta x}$ — don't invert it.