Geometry

Coordinate Geometry

Gradients, equations of lines, distance and parallel/perpendicular lines.

Key formulas

Gradient\[m=rac{y_2-y_1}{x_2-x_1}\]

Distance\[d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]

Parallel linessame gradient

Perpendicular lines\[m_1m_2=-1\]

Parallel lines have equal gradients; perpendicular lines have negative reciprocal gradients.

Find the equation of the line parallel to \(y=2x-6\) passing through \((2,5)\).

A parallel line has gradient 2, so \(y=2x+c\). Substitute \((2,5)\):

\[5=2(2)+c\Rightarrow c=1.\]

The equation is \(y=2x+1\).