2D Coordinate Geometry - Problem Solving

This page shows several examples for solving 2D coordinate geometry problems.

The most used formula about this is the distance formula. If you want to find the distance between \((x_1,y_1)\) and \((x_2,y_2)\), you may use the distance formula

\[\text{distance}=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}=\sqrt{(\triangle x)^2+(\triangle y)^2}.\]

The midpoint formula might come in handy, too. The midpoint of \((x_1,y_1)\) and \((x_2,y_2)\) is

\[\left(\frac{x_1+x_2}2,\frac{y_1+y_2}2\right).\]

The equation of lines is in the form of \(y=mx+n\) (where \(m\) is the slope and \(n\) is the y-intercept), or \(ax+by=c\).

What is the equation of the line passing through \((3,7)\) and \((5,4)?\)

---

Put the coordinates into \(y=mx+n\) and get \(7=3m+n\) and \(5=4m+n\).

Solve it and get \(m=-1.5\) and \(n=11.5,\) so the equation is \(y=-1.5x+11.5\) or \(3x+2y=23.\ _\square\)

The equation of conic sections are in the form of \(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\).