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

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

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Ron the Mouse travels 80 centimeters north and then 150 centimeters east.

Let \(d\) be the distance traveled by Ron the Mouse and \(s\) be the magnitude of the displacement of the mouse.

Find the value (in centimeters) of \(d-s.\)

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

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

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Let \(O=(0,0), P=(3,4), Q=(6,0)\) be the vertices of triangle \(OPQ.\) Point \(R\) inside the triangle is such that triangles \(OPR, PQR, OQR\) are of equal area.

The product of the coordinates of \(R\) is \(\text{__________}.\)

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 \((5,4)?\)

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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 \(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\).

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In the ellipse shown, let \(a\) and \(b\) be the lengths of the semi-major and semi-minor axes, respectively.

What is the value of \(j\) for which \(a^2 + b^2 = 19200?\)