GCSE Maths: Parallel and perpendicular lines (Higher)
Part of the GCSE Maths straight-line graphs and coordinates chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Parallel and perpendicular lines
Parallel lines go in the same direction and never meet, like the two rails of a straight railway track. They have exactly the same steepness, so parallel lines have equal gradients. In y = mx + c form, the lines y = 2x + 1, y = 2x − 3 and y = 2x are all parallel because every one has m = 2. Only the y-intercept (c) is different, which just slides the line up or down.
If an equation is not in the form y = mx + c, rearrange it first before comparing gradients. For example 2y = 4x + 6 becomes y = 2x + 3, so it is parallel to y = 2x + 1.
Finding a line parallel to another
To find the equation of a line parallel to y = 2x + 1 that passes through (3, 4): the gradient must be 2, so write y = 2x + c, substitute the point to get 4 = 6 + c, so c = −2. The line is y = 2x − 2. Keep the gradient, find a new c.
Perpendicular lines (Higher)
Two lines are perpendicular if they meet at a right angle (90°). Their gradients are linked: multiply the two gradients and you get −1. Equivalently, the perpendicular gradient is the negative reciprocal: flip the fraction upside down and change the sign.
Examples: a line with gradient 2 (which is 2/1) has a perpendicular line with gradient −1/2. A line with gradient −3/4 has a perpendicular line with gradient 4/3. Why does this work? Take a line with gradient 2: it goes 1 across and 2 up. Turn it a quarter turn anticlockwise: it now goes 2 across the opposite way and 1 up, which is “across −2, up 1”, a gradient of 1 ÷ (−2) = −1/2. A quarter turn swaps the rise and the run and changes one sign. Check: 2 × (−1/2) = −1.
Equation of a perpendicular line (Higher)
To find the line perpendicular to y = 2x + 1 through (4, 1): the perpendicular gradient is −½, so y = −½x + c. Substitute (4, 1): 1 = −2 + c, so c = 3. A perpendicular bisector of a line segment passes through its midpoint and is perpendicular to the segment.
Key ideas
- Parallel lines have equal gradients. Rearrange to y = mx + c to compare.
- A parallel line through a point: keep m, substitute the point to find the new c.
- Higher: perpendicular gradients multiply to −1: the perpendicular gradient is the negative reciprocal (flip and change sign).
- Higher: perpendicular bisector = through the midpoint, with the perpendicular gradient.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Rearrange the second equation into the form y = mx + c, then compare the gradients.
- Find the gradient of the first line. The equation is already y = mx + c.y = 3x − 4 has m = 3
- Rearrange the second equation. Divide every term by 3 so that y is on its own.3y = 9x + 2 → y = 3x + 23
- Compare the gradients. Parallel lines have the same gradient.3 = 3
- Write the conclusion. The lines are parallel.
Check. Both lines rise 3 for every 1 across. ✓
Watch out. You must rearrange first. In 3y = 9x + 2 the gradient is not 9.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = 2x + 1 has gradient m = 2
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = 2x + c
- Substitute the point (3, 4) to find c. The new line must go through this point.4 = 2 × 3 + c
4 = 6 + c
c = 4 − 6 = −2 - Write the equation. y = 2x − 2
Check. The gradient 2 matches the given line, and the point (3, 4) fits: 2 × 3 − 2 = 4. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = −12x + 3 has gradient m = −12
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = −12x + c
- Substitute the point (4, 5) to find c. The new line must go through this point.5 = (−12) × 4 + c
5 = −2 + c
c = 5 − (−2) = 7 - Write the equation. y = −12x + 7
Check. The gradient −12 matches the given line, and the point (4, 5) fits: (−12) × 4 + 7 = 5. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Key words. Two lines are perpendicular if they meet at a right angle (90°). The reciprocal of a number is 1 divided by it; the negative reciprocal also changes the sign.
Method. Perpendicular lines meet at a right angle. Their gradients multiply to give −1, so the perpendicular gradient is the negative reciprocal: flip the fraction and change the sign.
- Write the gradient as a fraction. A whole number is a fraction over 1. This makes it easy to flip.m = 23
- Flip the fraction upside down. The perpendicular gradient uses the reciprocal: the top and the bottom swap places.23 flips to 32
- Change the sign. The perpendicular gradient is the negative reciprocal: positive becomes negative, negative becomes positive.perpendicular gradient = −32
Check. Multiply the gradients: 23 × (−32) = −1. ✓
Watch out. You need both changes: flip and change sign. The reciprocal of 2 is ½, but the perpendicular gradient is −½.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Perpendicular lines have gradients that multiply to −1. Find the negative reciprocal of the gradient, then use the point to find c.
- Read the gradient of the given line. m is the number in front of x when the equation is y = mx + c.y = 2x + 1 has gradient 2
- Find the perpendicular gradient. Flip the fraction and change the sign (the negative reciprocal). Check: the two gradients must multiply to give −1.2 → −12
Check: 2 × (−12) = −1 - Put the new gradient in y = mx + c. Only c is unknown now.y = −12x + c
- Substitute the point (4, 1) to find c. The new line must pass through this point.1 = (−12) × 4 + c
1 = −2 + c
c = 1 − (−2) = 3 - Write the equation. y = −12x + 3
Check. The point fits: (−12) × 4 + 3 = 1 = 1. And 2 × (−12) = −1. ✓
Watch out. Flip and change the sign. Then substitute the point into the new equation, not the old one.
Key words. A bisector cuts something exactly in half. A perpendicular bisector of a line segment cuts it in half at 90°.
Method. A perpendicular bisector cuts a line segment in half at right angles. It goes through the midpoint and has the negative reciprocal gradient.
- Find the midpoint of AB. A bisector cuts the line in half, so it passes through the midpoint.M = ((0 + 4) ÷ 2, (1 + 3) ÷ 2) = (2, 2)
- Find the gradient of AB. Rise ÷ run.m = (3 − 1) ÷ (4 − 0) = 2 ÷ 4 = 12
- Find the perpendicular gradient. “Perpendicular” means at right angles to AB, so use the negative reciprocal.12 → −2
- Substitute the midpoint into y = mx + c. The bisector goes through the midpoint M.2 = (−2) × 2 + c
2 = −4 + c
c = 2 − (−4) = 6 - Write the equation. y = −2x + 6
Check. 12 × (−2) = −1, so it is perpendicular; and the midpoint fits: (−2) × 2 + 6 = 2 = 2. ✓
Watch out. Use the midpoint for the point, not one of the end points.
Watch out
Same gradient, not same c. Parallel lines have equal m; if c is also equal they are the same line. Always rearrange to y = mx + c first: the gradient of 2y = 6x + 1 is 3, not 6. For perpendicular lines you must flip and change the sign.Exercise 12I Parallel lines
Non-calculatorMethod. Parallel lines have the same gradient as the given line, so find m.
- Find the number multiplying x. y = 3x + 2 is in the form y = mx + c.m = 3
- Parallel lines have equal gradients. Same steepness, never meet.parallel gradient = 3
Check. Equal gradients mean the same slope. ✓
Watch out. Keep the sign of the gradient.
Method. Parallel lines have the same gradient as the given line, so find m.
- Find the number multiplying x. y = −x + 7 is in the form y = mx + c.m = −1
- Parallel lines have equal gradients. Same steepness, never meet.parallel gradient = −1
Check. Equal gradients mean the same slope. ✓
Watch out. Keep the sign of the gradient.
Method. Parallel lines have the same gradient as the given line, so find m.
- Find the number multiplying x. y = 12x − 1 is in the form y = mx + c.m = 12
- Parallel lines have equal gradients. Same steepness, never meet.parallel gradient = 12
Check. Equal gradients mean the same slope. ✓
Watch out. Keep the sign of the gradient.
Method. Parallel lines have the same gradient as the given line, so find m.
- Find the number multiplying x. y = −2x + 5 is in the form y = mx + c.m = −2
- Parallel lines have equal gradients. Same steepness, never meet.parallel gradient = −2
Check. Equal gradients mean the same slope. ✓
Watch out. Keep the sign of the gradient.
Method. Parallel lines have the same gradient as y = 2x + 1, which is 2. Check each line.
- Find the gradient of y = 2x + 1. The number multiplying x.gradient 2
- Check A to D. Write the gradient of each line.A: 2 ✓ B: 1 ✗ C: 2 ✓ D: −2 ✗
- Choose the lines with gradient 2. The y-intercepts do not matter.A and C
Check. A and C both go up 2 for each 1 across. ✓
Watch out. D has gradient −2: it goes down, so it is not parallel.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Read the y-intercept (c), find the gradient (m) with a triangle, then write y = mx + c.
- Find where the line crosses the y-axis. That value is c, the y-intercept.The line crosses the y-axis at (0, 1), so c = 1.
- Draw a gradient triangle. Choose two points on grid corners and count the run (across) and the rise (up or down).
- Work out the gradient. Gradient = rise ÷ run.rise = 3, run = 6
m = 3 ÷ 6 = 12 - Write the equation in the form y = mx + c. Put m and c into the formula.y = 12x + 1
Check. Test a point on the line, (−4, −1): y = 12 × (−4) + 1 = −1, which matches. ✓
Watch out. Do not mix up m and c: c is where the line hits the y-axis; m is the steepness. A negative gradient needs a minus sign.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Read the y-intercept (c), find the gradient (m) with a triangle, then write y = mx + c.
- Find where the line crosses the y-axis. That value is c, the y-intercept.The line crosses the y-axis at (0, −2), so c = −2.
- Draw a gradient triangle. Choose two points on grid corners and count the run (across) and the rise (up or down).
- Work out the gradient. Gradient = rise ÷ run.rise = 3, run = 6
m = 3 ÷ 6 = 12 - Write the equation in the form y = mx + c. Put m and c into the formula.y = 12x − 2
Check. Test a point on the line, (−4, −4): y = 12 × (−4) − 2 = −4, which matches. ✓
Watch out. Do not mix up m and c: c is where the line hits the y-axis; m is the steepness. A negative gradient needs a minus sign.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = 2x + 1 has gradient m = 2
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = 2x + c
- Substitute the point (3, 4) to find c. The new line must go through this point.4 = 2 × 3 + c
4 = 6 + c
c = 4 − 6 = −2 - Write the equation. y = 2x − 2
Check. The gradient 2 matches the given line, and the point (3, 4) fits: 2 × 3 − 2 = 4. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = −x + 4 has gradient m = −1
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = −x + c
- Substitute the point (2, 5) to find c. The new line must go through this point.5 = (−1) × 2 + c
5 = −2 + c
c = 5 − (−2) = 7 - Write the equation. y = −x + 7
Check. The gradient −1 matches the given line, and the point (2, 5) fits: −2 + 7 = 5. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = 12x + 3 has gradient m = 12
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = 12x + c
- Substitute the point (4, 0) to find c. The new line must go through this point.0 = 12 × 4 + c
0 = 2 + c
c = 0 − 2 = −2 - Write the equation. y = 12x − 2
Check. The gradient 12 matches the given line, and the point (4, 0) fits: 12 × 4 − 2 = 0. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = −3x has gradient m = −3
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = −3x + c
- Substitute the point (−1, 2) to find c. The new line must go through this point.2 = (−3) × (−1) + c
2 = 3 + c
c = 2 − 3 = −1 - Write the equation. y = −3x − 1
Check. The gradient −3 matches the given line, and the point (−1, 2) fits: (−3) × (−1) − 1 = 2. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Method. Two lines are parallel if they have the same gradient.
- Write the gradient of each line. Put each equation in the form y = mx + c. The gradient is m.y = 2x + 3: gradient 2
2y = 4x − 5 (divide by 2: y = 2x − 2.5): gradient 2 - Compare the gradients. Parallel lines have equal gradients: they slope the same way and never meet.2 = 2
- Decide. The lines are parallel.
Check. Draw both lines: parallel lines never cross. ✓
Watch out. The y-intercepts may be different: only the gradients need to match.
Method. Two lines are parallel if they have the same gradient.
- Write the gradient of each line. Put each equation in the form y = mx + c. The gradient is m.4x + 2y = 7 (subtract 4x, divide by 2: y = −2x + 3.5): gradient −2
y = −2x + 1: gradient −2 - Compare the gradients. Parallel lines have equal gradients: they slope the same way and never meet.−2 = −2
- Decide. The lines are parallel.
Check. Draw both lines: parallel lines never cross. ✓
Watch out. The y-intercepts may be different: only the gradients need to match.
Method. A quadrilateral with two pairs of parallel sides is a parallelogram. Parallel sides have equal gradients, so work out all four gradients.
- Draw a sketch. The sketch shows the shape.
- Gradient of AB and of DC. Rise ÷ run for each side, in the same direction.AB: (1 − (−1)) ÷ (2 − (−2)) = 12
DC: (4 − 2) ÷ (3 − (−1)) = 12 - Gradient of BC and of AD. The other pair of sides.BC: (4 − 1) ÷ (3 − 2) = 3
AD: (2 − (−1)) ÷ (−1 − (−2)) = 3 - Conclude. Equal gradients mean parallel sides.AB is parallel to DC, and BC is parallel to AD. ABCD has two pairs of parallel sides, so it is a parallelogram.
Check. The midpoints of the diagonals AC and BD are both (0.5, 1.5). ✓
Watch out. Compare gradients of opposite sides.
Method. Parallel lines have equal gradients. Write the gradient of each line and make them equal.
- Find the gradient of the first line. y = 3x + 1 is in the form y = mx + c.m = 3
- Find the gradient of the second line. In y = kx − 5 the gradient is k.m = k
- Make them equal. The lines are parallel.k = 3
Check. y = 3x + 1 and y = 3x − 5 both have gradient 3. ✓
Watch out. Only the gradients must match.
Method. Write the gradient of the second line in terms of k, then make it equal to 10.
- Find the gradient of y = 10x. m = 10
- Find the gradient of y = 2kx + 1. The number multiplying x is 2k.m = 2k
- Make them equal and solve. Divide both sides by 2.2k = 10, so k = 5
Check. 2 × 5 = 10. ✓
Watch out. The gradient is the whole number multiplying x, here 2k.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = 12x + 1 has gradient m = 12
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = 12x + c
- Substitute the point (4, 6) to find c. The new line must go through this point.6 = 12 × 4 + c
6 = 2 + c
c = 6 − 2 = 4 - Write the equation. y = 12x + 4
Check. The gradient 12 matches the given line, and the point (4, 6) fits: 12 × 4 + 4 = 6. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Method. Two lines with the same gradient go in the same direction and keep the same distance apart.
- Find the gradients. Both equations are in the form y = mx + c.y = 2x + 1 has m = 2 and y = 2x + 4 has m = 2
- Think about what equal gradients mean. For every 1 across, each line goes up 2.They stay the same vertical distance apart: 4 − 1 = 3.
- Conclude. Lines that stay the same distance apart can never meet.The lines are parallel and have different y-intercepts, so they never meet.
Check. Suppose they met: 2x + 1 = 2x + 4 would give 1 = 4, which is impossible. ✓
Watch out. “Parallel” is the key word.
Method. Find the gradient of each line. Lines with the same gradient are parallel.
- Write the gradient of each line. The number multiplying x. The line y = 3 has no x term, so its gradient is 0.A: 2, B: −1, C: 2, D: −1, E: 12, F: 0
- Group equal gradients. Equal gradients are parallel.Gradient 2: A and C. Gradient −1: B and D.
- Answer. E and F have no partner.The parallel pairs are A and C, and B and D.
Check. A and C both go up 2 for 1 across. B and D both go down 1 for 1 across. ✓
Watch out. Check the sign of each gradient.
Exercise 12J Perpendicular lines
Non-calculatorKey words. Two lines are perpendicular if they meet at a right angle (90°). The reciprocal of a number is 1 divided by it; the negative reciprocal also changes the sign.
Method. Perpendicular lines meet at a right angle. Their gradients multiply to give −1, so the perpendicular gradient is the negative reciprocal: flip the fraction and change the sign.
- Write the gradient as a fraction. A whole number is a fraction over 1. This makes it easy to flip.m = 2 = 21
- Flip the fraction upside down. The perpendicular gradient uses the reciprocal: the top and the bottom swap places.21 flips to 12
- Change the sign. The perpendicular gradient is the negative reciprocal: positive becomes negative, negative becomes positive.perpendicular gradient = −12
Check. Multiply the gradients: 2 × (−12) = −1. ✓
Watch out. You need both changes: flip and change sign. The reciprocal of 2 is ½, but the perpendicular gradient is −½.
Key words. Two lines are perpendicular if they meet at a right angle (90°). The reciprocal of a number is 1 divided by it; the negative reciprocal also changes the sign.
Method. Perpendicular lines meet at a right angle. Their gradients multiply to give −1, so the perpendicular gradient is the negative reciprocal: flip the fraction and change the sign.
- Write the gradient as a fraction. A whole number is a fraction over 1. This makes it easy to flip.m = −3 = -31
- Flip the fraction upside down. The perpendicular gradient uses the reciprocal: the top and the bottom swap places.-31 flips to 13
- Change the sign. The perpendicular gradient is the negative reciprocal: positive becomes negative, negative becomes positive.perpendicular gradient = 13
Check. Multiply the gradients: (−3) × 13 = −1. ✓
Watch out. You need both changes: flip and change sign. The reciprocal of 2 is ½, but the perpendicular gradient is −½.
Key words. Two lines are perpendicular if they meet at a right angle (90°). The reciprocal of a number is 1 divided by it; the negative reciprocal also changes the sign.
Method. Perpendicular lines meet at a right angle. Their gradients multiply to give −1, so the perpendicular gradient is the negative reciprocal: flip the fraction and change the sign.
- Write the gradient as a fraction. A whole number is a fraction over 1. This makes it easy to flip.m = 34
- Flip the fraction upside down. The perpendicular gradient uses the reciprocal: the top and the bottom swap places.34 flips to 43
- Change the sign. The perpendicular gradient is the negative reciprocal: positive becomes negative, negative becomes positive.perpendicular gradient = −43
Check. Multiply the gradients: 34 × (−43) = −1. ✓
Watch out. You need both changes: flip and change sign. The reciprocal of 2 is ½, but the perpendicular gradient is −½.
Key words. Two lines are perpendicular if they meet at a right angle (90°). The reciprocal of a number is 1 divided by it; the negative reciprocal also changes the sign.
Method. Perpendicular lines meet at a right angle. Their gradients multiply to give −1, so the perpendicular gradient is the negative reciprocal: flip the fraction and change the sign.
- Write the gradient as a fraction. A whole number is a fraction over 1. This makes it easy to flip.m = −25
- Flip the fraction upside down. The perpendicular gradient uses the reciprocal: the top and the bottom swap places.−25 flips to 52
- Change the sign. The perpendicular gradient is the negative reciprocal: positive becomes negative, negative becomes positive.perpendicular gradient = 52
Check. Multiply the gradients: (−25) × 52 = −1. ✓
Watch out. You need both changes: flip and change sign. The reciprocal of 2 is ½, but the perpendicular gradient is −½.
Method. Two lines are perpendicular if the product of their gradients is −1.
- Write down the two gradients. Both equations are in the form y = mx + c, so the gradient is the number multiplying x.first line: gradient 2
second line: gradient −12 - Multiply the gradients. Lines are perpendicular (at right angles) when the gradients multiply to give −1.2 × (−12) = −1
- Decide. The product is −1, so the lines are perpendicular.
Check. −12 is the negative reciprocal of 2: flip and change sign. ✓
Watch out. Both gradients being negative (or both positive) can never give −1: perpendicular gradients have opposite signs.
Method. Two lines are perpendicular if the product of their gradients is −1.
- Write down the two gradients. Both equations are in the form y = mx + c, so the gradient is the number multiplying x.first line: gradient 3
second line: gradient 3 - Multiply the gradients. Lines are perpendicular (at right angles) when the gradients multiply to give −1.3 × 3 = 9
- Decide. The product is 9, which is not −1, so the lines are not perpendicular.
Check. The perpendicular gradient to 3 would be −13, not 3.
Watch out. Both gradients being negative (or both positive) can never give −1: perpendicular gradients have opposite signs.
Method. Two lines are perpendicular if the product of their gradients is −1.
- Write down the two gradients. Both equations are in the form y = mx + c, so the gradient is the number multiplying x.first line: gradient 34
second line: gradient −43 - Multiply the gradients. Lines are perpendicular (at right angles) when the gradients multiply to give −1.34 × (−43) = −1
- Decide. The product is −1, so the lines are perpendicular.
Check. −43 is the negative reciprocal of 34: flip and change sign. ✓
Watch out. Both gradients being negative (or both positive) can never give −1: perpendicular gradients have opposite signs.
Method. Two lines are perpendicular if the product of their gradients is −1.
- Write down the two gradients. Both equations are in the form y = mx + c, so the gradient is the number multiplying x.first line: gradient −2
second line: gradient 2 - Multiply the gradients. Lines are perpendicular (at right angles) when the gradients multiply to give −1.(−2) × 2 = −4
- Decide. The product is −4, which is not −1, so the lines are not perpendicular.
Check. The perpendicular gradient to −2 would be 12, not 2.
Watch out. Both gradients being negative (or both positive) can never give −1: perpendicular gradients have opposite signs.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Perpendicular lines have gradients that multiply to −1. Find the negative reciprocal of the gradient, then use the point to find c.
- Read the gradient of the given line. m is the number in front of x when the equation is y = mx + c.y = 2x + 3 has gradient 2
- Find the perpendicular gradient. Flip the fraction and change the sign (the negative reciprocal). Check: the two gradients must multiply to give −1.2 → −12
Check: 2 × (−12) = −1 - Put the new gradient in y = mx + c. Only c is unknown now.y = −12x + c
- Substitute the point (4, 1) to find c. The new line must pass through this point.1 = (−12) × 4 + c
1 = −2 + c
c = 1 − (−2) = 3 - Write the equation. y = −12x + 3
Check. The point fits: (−12) × 4 + 3 = 1 = 1. And 2 × (−12) = −1. ✓
Watch out. Flip and change the sign. Then substitute the point into the new equation, not the old one.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Perpendicular lines have gradients that multiply to −1. Find the negative reciprocal of the gradient, then use the point to find c.
- Read the gradient of the given line. m is the number in front of x when the equation is y = mx + c.y = 12x + 1 has gradient 12
- Find the perpendicular gradient. Flip the fraction and change the sign (the negative reciprocal). Check: the two gradients must multiply to give −1.12 → −2
Check: 12 × (−2) = −1 - Put the new gradient in y = mx + c. Only c is unknown now.y = −2x + c
- Substitute the point (2, 5) to find c. The new line must pass through this point.5 = (−2) × 2 + c
5 = −4 + c
c = 5 − (−4) = 9 - Write the equation. y = −2x + 9
Check. The point fits: (−2) × 2 + 9 = 5 = 5. And 12 × (−2) = −1. ✓
Watch out. Flip and change the sign. Then substitute the point into the new equation, not the old one.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Perpendicular lines have gradients that multiply to −1. Find the negative reciprocal of the gradient, then use the point to find c.
- Read the gradient of the given line. m is the number in front of x when the equation is y = mx + c.y = −3x + 2 has gradient −3
- Find the perpendicular gradient. Flip the fraction and change the sign (the negative reciprocal). Check: the two gradients must multiply to give −1.−3 → 13
Check: (−3) × 13 = −1 - Put the new gradient in y = mx + c. Only c is unknown now.y = 13x + c
- Substitute the point (6, 0) to find c. The new line must pass through this point.0 = 13 × 6 + c
0 = 2 + c
c = 0 − 2 = −2 - Write the equation. y = 13x − 2
Check. The point fits: 13 × 6 − 2 = 0 = 0. And (−3) × 13 = −1. ✓
Watch out. Flip and change the sign. Then substitute the point into the new equation, not the old one.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Perpendicular lines have gradients that multiply to −1. Find the negative reciprocal of the gradient, then use the point to find c.
- Read the gradient of the given line. m is the number in front of x when the equation is y = mx + c.y = 4x − 1 has gradient 4
- Find the perpendicular gradient. Flip the fraction and change the sign (the negative reciprocal). Check: the two gradients must multiply to give −1.4 → −14
Check: 4 × (−14) = −1 - Put the new gradient in y = mx + c. Only c is unknown now.y = −14x + c
- Substitute the point (8, 3) to find c. The new line must pass through this point.3 = (−14) × 8 + c
3 = −2 + c
c = 3 − (−2) = 5 - Write the equation. y = −14x + 5
Check. The point fits: (−14) × 8 + 5 = 3 = 3. And 4 × (−14) = −1. ✓
Watch out. Flip and change the sign. Then substitute the point into the new equation, not the old one.
Key words. A bisector cuts something exactly in half. A perpendicular bisector of a line segment cuts it in half at 90°.
Method. A perpendicular bisector cuts a line segment in half at right angles. It goes through the midpoint and has the negative reciprocal gradient.
- Find the midpoint of AB. A bisector cuts the line in half, so it passes through the midpoint.M = ((0 + 4) ÷ 2, (1 + 3) ÷ 2) = (2, 2)
- Find the gradient of AB. Rise ÷ run.m = (3 − 1) ÷ (4 − 0) = 2 ÷ 4 = 12
- Find the perpendicular gradient. “Perpendicular” means at right angles to AB, so use the negative reciprocal.12 → −2
- Substitute the midpoint into y = mx + c. The bisector goes through the midpoint M.2 = (−2) × 2 + c
2 = −4 + c
c = 2 − (−4) = 6 - Write the equation. y = −2x + 6
Check. 12 × (−2) = −1, so it is perpendicular; and the midpoint fits: (−2) × 2 + 6 = 2 = 2. ✓
Watch out. Use the midpoint for the point, not one of the end points.
Key words. A bisector cuts something exactly in half. A perpendicular bisector of a line segment cuts it in half at 90°.
Method. A perpendicular bisector cuts a line segment in half at right angles. It goes through the midpoint and has the negative reciprocal gradient.
- Find the midpoint of AB. A bisector cuts the line in half, so it passes through the midpoint.M = ((−2 + 2) ÷ 2, (3 + (−3)) ÷ 2) = (0, 0)
- Find the gradient of AB. Rise ÷ run.m = (−3 − 3) ÷ (2 − (−2)) = −6 ÷ 4 = −32
- Find the perpendicular gradient. “Perpendicular” means at right angles to AB, so use the negative reciprocal.−32 → 23
- Substitute the midpoint into y = mx + c. The bisector goes through the midpoint M.0 = 23 × 0 + c
0 = 0 + c
c = 0 − 0 = 0 - Write the equation. y = 23x
Check. (−32) × 23 = −1, so it is perpendicular; and the midpoint fits: 23 × 0 = 0 = 0. ✓
Watch out. Use the midpoint for the point, not one of the end points.
Key words. A bisector cuts something exactly in half. A perpendicular bisector of a line segment cuts it in half at 90°.
Method. A perpendicular bisector cuts a line segment in half at right angles. It goes through the midpoint and has the negative reciprocal gradient.
- Find the midpoint of AB. A bisector cuts the line in half, so it passes through the midpoint.M = ((1 + 5) ÷ 2, (0 + 2) ÷ 2) = (3, 1)
- Find the gradient of AB. Rise ÷ run.m = (2 − 0) ÷ (5 − 1) = 2 ÷ 4 = 12
- Find the perpendicular gradient. “Perpendicular” means at right angles to AB, so use the negative reciprocal.12 → −2
- Substitute the midpoint into y = mx + c. The bisector goes through the midpoint M.1 = (−2) × 3 + c
1 = −6 + c
c = 1 − (−6) = 7 - Write the equation. y = −2x + 7
Check. 12 × (−2) = −1, so it is perpendicular; and the midpoint fits: (−2) × 3 + 7 = 1 = 1. ✓
Watch out. Use the midpoint for the point, not one of the end points.
Method. Rearrange both equations into the form y = mx + c, find the gradients and multiply them.
- Rearrange 2x + 3y = 6. Subtract 2x from both sides and divide by 3.3y = −2x + 6, so y = −23x + 2; gradient −23
- Rearrange 3x − 2y = 4. Subtract 3x from both sides, then divide by −2.−2y = −3x + 4, so y = 32x − 2; gradient 32
- Multiply the gradients. Perpendicular lines give −1.−23 × 32 = −1
- Conclude. The product is −1, so the lines are perpendicular.
Check. The gradient 32 is the negative reciprocal of −23. ✓
Watch out. When you divide by −2 all the signs change.
Method. Two sides are perpendicular if their gradients multiply to −1. Find the gradients of AB and BC.
- Draw a sketch. The sketch shows the triangle.
- Gradient of AB. Rise ÷ run.(3 − 1) ÷ (5 − 1) = 2 ÷ 4 = 12
- Gradient of BC. (7 − 3) ÷ (3 − 5) = 4 ÷ (−2) = −2
- Multiply. 12 × (−2) = −1
- Conclude. The gradients multiply to −1, so AB is perpendicular to BC.There is a right angle at B, so ABC is a right-angled triangle.
Check. Pythagoras: AB² = 20, BC² = 20, AC² = 40 and 20 + 20 = 40. ✓
Watch out. Choose the two sides that meet at the suspected right angle.
Method. A rectangle has opposite sides parallel and adjacent sides perpendicular. Check the four gradients.
- Draw a sketch.
- Find the four gradients. Rise ÷ run for each side in turn.AB: 12, BC: −2, CD: 12, DA: −2
- Check for parallel sides. Equal gradients mean parallel.AB and CD both 12; BC and DA both −2
- Check for right angles. Gradients that multiply to −1 are perpendicular.12 × (−2) = −1, so AB ⊥ BC
- Conclude. A parallelogram with a right angle is a rectangle.Opposite sides are parallel and adjacent sides are perpendicular, so ABCD is a rectangle.
Check. The diagonals AC and BD both have midpoint (1, 3). ✓
Watch out. Show both: parallel opposite sides and a right angle.
Method. Rearrange 3y + 2x = 5 to find its gradient, then use the negative reciprocal.
- Rearrange. Subtract 2x and divide by 3.3y = −2x + 5, so y = −23x + 53
- Find the gradient. −23
- Find the perpendicular gradient. Flip and change the sign.32
- Compare with y = kx + 1. The gradient is k.k = 32
Check. −23 × 32 = −1. ✓
Watch out. Rearrange first.
Method. Find the gradient of AB, then the perpendicular gradient, then use the point P to find c.
- Gradient of AB. A(0, 1), B(4, 3).(3 − 1) ÷ (4 − 0) = 12
- Perpendicular gradient. Flip and change sign.−2
- Substitute P(2, 5). y = −2x + c.5 = −4 + c, so c = 9
- Write the equation. y = −2x + 9
Check. Test P: −2 × 2 + 9 = 5. ✓
Watch out. Use the point P, not the midpoint.
Method. A horizontal line goes along; a vertical line goes up. Use the shape of the lines.
- Describe a line with gradient 0. Gradient 0 means no rise.It is horizontal, like y = 3.
- Describe a vertical line. Its run is 0, so its gradient is undefined.It is straight up and down, like x = 2.
- Compare. The corner of a window frame is a right angle between a horizontal and a vertical edge.A horizontal line and a vertical line always meet at 90°, so they are perpendicular.
Check. y = 3 and x = 2 meet at (2, 3) and form a right angle. ✓
Watch out. The product-of-gradients rule does not apply here because a vertical line has no gradient.
Key words. A tangent is a straight line that touches a circle at exactly one point, and it is perpendicular to the radius at that point.
Method. A tangent to a circle is perpendicular to the radius at the point of contact. Find the gradient of the radius, then the tangent.
- Gradient of the radius CP. C(2, 1) and P(6, 4).(4 − 1) ÷ (6 − 2) = 34
- Gradient of the tangent. Perpendicular to the radius: flip and change sign.−43
- Find c. Substitute P(6, 4) into y = −43x + c.4 = −8 + c, so c = 12
- Write the equation. y = −43x + 12
Check. Test P: −43 × 6 + 12 = −8 + 12 = 4. ✓
Watch out. Use P (on the circle), not the centre, to find c.