Parallel and perpendicular lines
| English | Chinese | Pinyin |
|---|---|---|
| parallel | 平行 | píng xíng |
| perpendicular | 垂直 | chuí zhí |
| negative reciprocal | 负倒数 | fù dào shǔ |
| perpendicular bisector | 垂直平分线 | chuí zhí píng fēn xiàn |
Railway tracks and city grids
- Railway tracks run parallel 平行 — the same gradient, never meeting, stretching to the horizon.
- Streets in a planned city like New York or Barcelona cross at right angles, forming a grid.
- Both ideas — parallel and perpendicular 垂直 — come down to one thing: the gradient.
Parallel lines
- Parallel lines never meet, no matter how far you extend them.
- Two lines are parallel if and only if they have the same gradient.
- $y = 4x - 1$ and $y = 4x + 7$ are parallel (both have $m = 4$).
Find a parallel line. The line parallel to $y = 4x - 1$ through $(1, -3)$: same gradient $4$, so $-3 = 4(1) + c \Rightarrow c = -7$. The equation is $y = 4x - 7$.
Parallel lines share a gradient; perpendicular gradients multiply to $-1$
Parallel & perpendicular
y = ax + b
Parallel lines share a gradient; perpendicular gradients multiply to −1.
A line is parallel to y = 4x − 1 and passes through (1, −3). In y = 4x + c, what is c?
Parallel → same gradient 4. −3 = 4(1) + c, so c = −7.
Which line is parallel to y = −2x + 5?
Parallel lines have the same gradient. y = −2x − 3 has gradient −2, the same as y = −2x + 5.
Perpendicular lines (Extended)
- Two lines are perpendicular if they meet at a right angle ($90°$).
- Their gradients multiply to $-1$:
- In words: flip the fraction and change the sign (the negative reciprocal 负倒数).
- Gradient $\dfrac{3}{2}$ → perpendicular gradient $-\dfrac{2}{3}$.
Parallel (cyan): same gradient. Perpendicular (rose + violet): gradients multiply to $-1$.
A line has gradient 3/2. The gradient of a line perpendicular to it is:
Flip and change sign: perpendicular gradient = −1 ÷ (3/2) = −2/3.
Two perpendicular lines have gradients that multiply to −1.
m₁ × m₂ = −1 is the defining property of perpendicular gradients.
If two lines are perpendicular, then m₁ × m₂ = ______.
The product of perpendicular gradients is always −1.
Worked example — perpendicular line
- Find the line perpendicular to $y = 2x + 3$ through $(4, 1)$.
- Original gradient $m_1 = 2$, so perpendicular gradient $m_2 = -\dfrac{1}{2}$.
- Substitute: $1 = -\dfrac{1}{2}(4) + c \Rightarrow 1 = -2 + c \Rightarrow c = 3$.
- The perpendicular line is $y = -\dfrac{1}{2}x + 3$.
Don't forget both steps. The perpendicular gradient is flip AND change sign. Gradient $3$ → perpendicular $-\dfrac{1}{3}$ (not $\dfrac{1}{3}$, which would just be the reciprocal).
A line has gradient 5. What is the gradient of a line perpendicular to it?
Perpendicular gradient = −1/5 = −0.2.
Perpendicular bisector 垂直平分线 (Extended)
- The perpendicular bisector of a segment cuts it in half at a right angle.
- Method (three steps):
- 1. Find the midpoint of the segment.
- 2. Find the perpendicular gradient ($-\dfrac{1}{m}$).
- 3. Use the perpendicular gradient and the midpoint to write the equation.
Match each gradient pair (m₁, m₂) to the relationship between the lines.
2/3 × (−3/2) = −1 → perpendicular. 4 = 4 → parallel. 1 × (−1) = −1 → perpendicular. 5 × (−5) = −25 ≠ −1 and 5 ≠ −5 → neither.
Put the steps for finding the perpendicular bisector of (0, 0) and (6, 2) in order.
First find the midpoint, then the segment gradient, then flip-and-sign for the perpendicular gradient, then substitute into y = mx + c.
City planners and gradient rules
- Manhattan's grid was planned in 1811 with avenues running north-south and streets east-west — all perpendicular.
- In contrast, old European cities have streets that meet at every angle — no gradient rule at all.
- Engineers use perpendicular gradients to design road junctions, railway crossings, and building foundations.
You've got it
- parallel → same gradient ($m_1 = m_2$)
- perpendicular → gradients multiply to $-1$ ($m_2 = -\dfrac{1}{m_1}$, "flip and change sign")
- perpendicular bisector = perpendicular gradient through the midpoint
- parallel to $y = 4x - 1$ through $(1, -3)$ → $y = 4x - 7$