From simple gradients to perpendicular intersections. Master the mechanics of lines.
The fundamental measure of steepness. Rise over run.
Find the slope of a line passing through \((2, 3)\) and \((5, 12)\).
\(m = \frac{12 - 3}{5 - 2} = \frac{9}{3} = 3\)
Result: 3.
Parallel lines never meet because they have identical slopes and different y-intercepts.
Which line is parallel to \(y = 4x + 7\)?
Look for any line with \(m = 4\). For instance, \(y = 4x - 10\) or \(4x - y = 5\).
Perpendicular slopes are negative reciprocals. Their product is always -1.
Rotation of a line by \(90^\circ\) swaps the horizontal and vertical components while negating one of them.
If a slope is \(\frac{\Delta y}{\Delta x}\), after \(90^\circ\) rotation, the new "Rise" becomes the old "Run", and the new "Run" becomes the negative of the old "Rise".
\(m_{new} = \frac{\Delta x}{-\Delta y} = -\frac{1}{m_{old}}\)