Lesson Guide
Explanation
Parallel lines never meet, because they rise at the same rate. So two lines are parallel exactly when their gradients are equal. Is y equals three x plus two parallel to six x minus two y equals five? Rearrange the second one: two y equals six x minus five, so y equals three x minus two and a half. Both gradients are three, so the lines are parallel. Their y-intercepts are different, so they are two separate lines.
Your Turn
Is y = −2x + 1 parallel to 4x + 2y = 9? Check: y = −2x + 4.5, gradient −2, so yes.
Watch Out
Compare gradients only after making y the subject. 6x − 2y = 5 does not have gradient 6.
Explanation
Perpendicular lines meet at a right angle. Take the step triangle for a gradient of two thirds, a run of three and a rise of two, and turn it a quarter turn. It becomes a run of two backwards and a rise of three: a gradient of negative three halves. So the perpendicular gradient is the negative reciprocal: flip the fraction and change its sign. The quick test is that the two gradients multiply to negative one. Two thirds times negative three halves is negative one.
Your Turn
Find the gradient perpendicular to −4, and to \(\dfrac{5}{2}\). Check: \(\dfrac{1}{4}\) and −\(\dfrac{2}{5}\).
Watch Out
Do both: flip AND change the sign. −\(\dfrac{2}{3}\) (sign only) and \(\dfrac{3}{2}\) (flip only) are both wrong.
Explanation
To find the line through two, three that is parallel to y equals four x minus one, keep the gradient, four, and use the point. y minus three equals four times, x minus two. Expand: y minus three equals four x minus eight, so y equals four x minus five. Check: four times two, minus five, is three. The new line has the same gradient but a different y-intercept.
Your Turn
Parallel to y = −x + 7, through (3, 1). Check: y = −x + 4.
Watch Out
Do not copy the old c. A parallel line through a new point has a new y-intercept.
Explanation
For the line through four, one that is perpendicular to y equals two x plus five: the given gradient is two, so the perpendicular gradient is negative one half. Then y minus one equals negative one half times, x minus four. Expand: y minus one equals negative one half x, plus two, so y equals negative one half x, plus three. Check both facts: two times negative one half is negative one, and the point four, one fits.
Your Turn
Perpendicular to y = −3x, through (3, 2). Check: m = \(\dfrac{1}{3}\), so y = \(\dfrac{1}{3}\)x + 1.
Watch Out
Put the NEW gradient into the formula. Using 2 again gives a parallel line, not a perpendicular one.
Explanation
Gradients can prove what kind of quadrilateral you have. Take A at zero, one, B at four, three, C at three, five and D at negative one, three. A B has gradient one half and so does C D, so they are parallel. B C and D A both have gradient negative two, so they are parallel too. Two pairs of parallel sides make a parallelogram. And one half times negative two is negative one, so A B is perpendicular to B C. A parallelogram with a right angle is a rectangle. Quiz and exercises now.
Your Turn
P(0, 0), Q(3, 1), R(4, 4), S(1, 3). Check: gradients \(\dfrac{1}{3}\), 3, \(\dfrac{1}{3}\), 3, so a parallelogram, but \(\dfrac{1}{3}\) × 3 = 1, not −1, so not a rectangle.
Watch Out
Parallel sides alone only prove a parallelogram. A rectangle also needs a right angle: adjacent gradients multiplying to −1.
📌 Key Points — this lesson in a nutshell
- Parallel lines have equal gradients: m1 = m2. Make y the subject before you compare.
- Perpendicular lines have gradients that multiply to −1: m1 × m2 = −1. The perpendicular gradient is the negative reciprocal: flip it AND change its sign.
- A horizontal line and a vertical line are perpendicular too, even though the vertical one has no gradient.
- For a line through a point parallel or perpendicular to another, find the new gradient first, then use y − y1 = m(x − x1).
- Both pairs of opposite sides parallel: a parallelogram. Add a right angle (adjacent gradients multiply to −1): a rectangle.
Step 1: Introduction
Step 1 of 5
Was this lesson helpful? Rate this Course
Discussion
Loading comments...