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Coordinate Geometry Foundations

Lesson · Beginner

Geometry: Coordinate Geometry

Slope, Line Equation, and Distance

Line through A(x_1,y_1) and B(x_2,y_2)
mAB=y2−y1x2−x1m_{AB}=\frac{y_2-y_1}{x_2-x_1}
y−y1=m(x−x1)y-y_1=m(x-x_1)
AB=(x2−x1)2+(y2−y1)2AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
Perpendicular lines with slopes m_1 and m_2
Perpendicular lines:m1m2=−1\text{Perpendicular lines:}\qquad m_1m_2=-1

Distance from a Point to a Line

Distance from point P to line l
For P(x0,y0) and ℓ:ax+by+c=0\text{For }P(x_0,y_0)\text{ and }\ell:ax+by+c=0
d(P,ℓ)=∣ax0+by0+c∣a2+b2d(P,\ell)=\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}

Distance Between Two Parallel Lines

Distance between two parallel lines
ℓ1:ax+by+c1=0,ℓ2:ax+by+c2=0\ell_1:ax+by+c_1=0,\qquad \ell_2:ax+by+c_2=0
d(ℓ1,ℓ2)=∣c1−c2∣a2+b2d(\ell_1,\ell_2)=\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}

Distance from a Point to a Plane

Distance from point P to plane alpha
For P(x0,y0,z0) and α:ax+by+cz+d=0\text{For }P(x_0,y_0,z_0)\text{ and }\alpha:ax+by+cz+d=0
d(P,α)=∣ax0+by0+cz0+d∣a2+b2+c2d(P,\alpha)=\frac{|ax_0+by_0+cz_0+d|}{\sqrt{a^2+b^2+c^2}}
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