GCSE · Maths · OCR · Spec J560
Parallel and perpendicular lines
Take a line and turn it through a right angle. Watch what happens to its rise and run. That one swap is the secret behind this whole topic.
Maths · Coordinate geometry
Parallel copy PR: gradient ½ again · Quarter-turn line OG: gradient −2
The quarter-turn swaps rise and run and reverses one of them: ½ × (−2) = −1
Same scale on both axes: 1 unit across is the same length as 1 unit up.
Worked example: a perpendicular line through a point
Problem
Find the equation of the line perpendicular to y = −2x + 10 that passes through (4, 25).
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
What you need to know
- Parallel lines have the same gradient.
- Perpendicular gradients are negative reciprocals, so they multiply to −1. For example, 2 and −½.
Have a goZed insists lines with gradients 3 and −⅓ can't be perpendicular because 'the numbers don't match'. Is Zed right?
No. 3 × (−⅓) = −1, so they are perpendicular.
Perpendicular gradients are not equal. They are negative reciprocals, and the product being −1 is the check.
- Gradient of a segment from its end points: change in y divided by change in x, taking care with negative values.
Have a goWithout calculating: a segment goes from (0, 9) to (3, 0). Will its gradient be positive or negative?
Negative: −9 ÷ 3 = −3.
y falls by 9 while x rises by 3, so change in y is negative and change in x is positive. A negative divided by a positive is negative.
- To find a perpendicular line through a point: take the negative reciprocal, write y = mx + c, then substitute the point to find c.
- A perpendicular bisector cuts a segment exactly in half at a right angle, so it passes through the midpoint.
- You can construct one with compasses and a ruler, but a drawn line is less accurate than working algebraically.
- If the scales on the axes are not 1 : 1, perpendicular lines do not look perpendicular.
Have a goMia draws two perpendicular lines but her corner looks nothing like a right angle. Her y-axis goes up in 5s and her x-axis in 1s. Has she gone wrong?
Not necessarily. Unequal scales stop perpendicular lines looking perpendicular.
The picture is stretched, not the maths. Make both axes go up in the same steps and the lines look perpendicular.
- A quadrilateral is a parallelogram if both pairs of opposite sides have equal gradients. It's a rectangle if adjacent sides are also perpendicular.
- A parallelogram with perpendicular diagonals is a rhombus. A rectangle with perpendicular diagonals is a square.
Have a goA parallelogram's diagonals have gradients 6 and −⅙. Which shape is it for sure?
A rhombus.
6 × (−⅙) = −1, so the diagonals are perpendicular, and a parallelogram with perpendicular diagonals is a rhombus.
- A kite's diagonals meet at a right angle but don't bisect each other, so a kite is not a rhombus.
The big picture
Parallel lines share a gradient. Perpendicular lines have gradients that are negative reciprocals, so they multiply to make −1. With those two facts you can build perpendicular lines and bisectors, and identify shapes on coordinate axes.
Key points
Worked example
Problem
Show that the line through P(2, 1) and Q(7, 4) is perpendicular to the line y = −(5/3)x + 2.
⚠ Watch out
Slipping on signs when finding a gradient. For (−2, 3) and (4, −3), working out −3 − 3 on top but −2 − 4 underneath gives +1 instead of −1. Subtract in the same order top and bottom.
Memory hook
Parallel: same. Perpendicular: flip it, switch it, check it makes −1.
Check yourself
Line L has gradient 3/4. What are the gradients of a parallel line and a perpendicular line? Answer: 3/4 and −4/3, since 3/4 × (−4/3) = −1.
Flashcards
(14)What do parallel lines have in common, gradient-wise?
How are the gradients of two perpendicular lines related?
What is the negative reciprocal of ⅔?
How do you find a segment's gradient from its end points?
The four stages for a line perpendicular to a given line through a given point?
What is a perpendicular bisector?
How do you construct a perpendicular bisector with compasses and a ruler?
Compass construction or algebra: which gives the more accurate bisector?
What happens to perpendicular lines drawn on axes that aren't 1 : 1?
When does a quadrilateral count as a parallelogram, using gradients?
A parallelogram becomes a rectangle when…?
What do the diagonals of a rhombus do?
A rectangle with perpendicular diagonals is a…?
Why is a kite not a rhombus, even though its diagonals are perpendicular?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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