Parallel and perpendicular lines
Parallel lines
- Parallel lines never meet, so they have the same gradient.
- Example: the line parallel to $y = 4x - 1$ through $(1, -3)$ also has gradient $4$.
- $-3 = 4(1) + c \Rightarrow c = -7$, so $y = 4x - 7$.
Practice
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.
Perpendicular lines (Extended)
- Two lines are perpendicular if they meet at a right angle. Their gradients multiply to $-1$:
$$m_1 \times m_2 = -1 \qquad\Rightarrow\qquad m_2 = -\frac{1}{m_1}$$
- In words: flip the fraction and change the sign.
- Example: gradient $\tfrac32$ → perpendicular gradient $-\tfrac23$.
Practice
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.
Practice
Two perpendicular lines have gradients that multiply to −1.
m₁ × m₂ = −1 is the test for perpendicular lines.
Perpendicular bisector (Extended)
- It cuts a segment in half at a right angle.
- Method: find the midpoint, then use the perpendicular gradient through that midpoint.
You've got it
Key idea
- parallel → same gradient
- perpendicular → gradients multiply to $-1$ (flip and change sign)
- perpendicular bisector = perpendicular gradient through the midpoint