Lines

Parallel and Perpendicular Lines

Build parallel and perpendicular lines through a point by controlling their slopes exactly.

Example

Use slope relationships to build parallel and perpendicular lines.

highlighted = computed this step

Step 1 — Plot the lines

Plot the related lines through P.

Coordinate plot\text{Coordinate plot}
Parallel and perpendicular plotGiven line, parallel line, and perpendicular line through P(0, 3).givenparallelperpendicularP

Step 2 — Given line and point

Read the given slope and point.

given: y=2x+1P=(0,3)\text{given: } y= \hl{2} x + \hl{1} \quad P=( \hl{0} , \hl{3} )

Step 3 — Parallel line

Keep the same slope for the parallel line.

m∥=2y=2x+3m_{\parallel}= \hl{2} \quad y= \hl{2} x + \hl{3}

Step 4 — Perpendicular slope

Use the negative reciprocal for perpendicular slope.

m⊥=−1/2=−12m_{\perp}= -1 / 2 = \hlmath{\frac{-1}{2}}

Step 5 — Perpendicular line

Write the perpendicular line through P.

y=−12x+3y= \hlmath{\frac{-1}{2}} x + \hl{3}

Step 6 — Check slopes

Check the perpendicular slope product.

2⋅−12=-12 \cdot \frac{-1}{2} = \hl{-1}
parallel-perpendicular-lines Parallel slopes match; perpendicular slopes multiply to -1.