IGCSE Mathematics: Parallel and Perpendicular Lines Practice Questions
Parallel lines have equal gradients. Perpendicular lines have gradients whose product is minus 1, so each gradient is the negative reciprocal of the other.
Topic 3.5 of Cambridge IGCSE Mathematics 0580 is short and highly predictable, which makes it reliable marks. The negative reciprocal is where errors happen, because both the flip and the sign change must be applied. The questions below force both steps.
What you need to know for Parallel and Perpendicular Lines
- Parallel linesTwo lines are parallel if and only if their gradients are equal. Their y-intercepts differ, otherwise they would be the same line.
- Perpendicular linesTwo lines are perpendicular if the product of their gradients is minus 1. Each gradient is the negative reciprocal of the other.
- Finding a negative reciprocalTurn the fraction upside down and change the sign. The negative reciprocal of 2, which is 2 over 1, is minus one half.
- Line through a given pointWrite y = mx + c with the required gradient, substitute the coordinates of the point, and solve for c.
- Perpendicular bisectorPasses through the midpoint of a line segment at right angles to it. Find the midpoint, find the gradient of the segment, take its negative reciprocal, then use the point and gradient.
- Checking perpendicularityMultiply the two gradients. If the product is minus 1 the lines are perpendicular, whatever they look like on a sketch.
IGCSE Mathematics Parallel and Perpendicular Lines questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Find the equation of the line parallel to y = 4x minus 3 that passes through the point (1, 9).
Show the worked answer
- Parallel lines have equal gradients, so the new line also has gradient 4.
- Write the partial equation: y = 4x + c.
- Substitute the point (1, 9): 9 = 4 x 1 + c, so 9 = 4 + c.
- c = 5, giving the equation y = 4x + 5.
Find the equation of the line perpendicular to y = 2x + 1 that passes through the point (4, 3).
Show the worked answer
- The gradient of the given line is 2. For a perpendicular line, take the negative reciprocal.
- The reciprocal of 2 is one half, and changing the sign gives minus one half. Check: 2 multiplied by minus one half = minus 1.
- Write the partial equation: y = minus one half x + c.
- Substitute (4, 3): 3 = minus one half times 4 + c, so 3 = minus 2 + c and c = 5. The equation is y = minus one half x + 5.
Determine whether the lines y = 3x minus 1 and y = minus x divided by 3, plus 4, are perpendicular. Justify your answer.
Show the worked answer
- Read the gradient of the first line: m1 = 3.
- Read the gradient of the second line: m2 = minus one third.
- Multiply the gradients: 3 multiplied by minus one third = minus 1.
- The product is minus 1, so the two lines are perpendicular. The y-intercepts are irrelevant to this test.
A is the point (1, 2) and B is the point (7, 10). Find the equation of the perpendicular bisector of AB.
Show the worked answer
- Find the midpoint of AB: x = (1 + 7) divided by 2 = 4, and y = (2 + 10) divided by 2 = 6. The midpoint is (4, 6).
- Find the gradient of AB: (10 minus 2) divided by (7 minus 1) = 8 divided by 6 = four thirds.
- The perpendicular gradient is the negative reciprocal of four thirds, which is minus three quarters. Check: four thirds times minus three quarters = minus 1.
- Write y = minus three quarters x + c and substitute the midpoint (4, 6): 6 = minus three quarters times 4 + c, so 6 = minus 3 + c.
- c = 9, giving the perpendicular bisector y = minus three quarters x + 9.
Common mistakes in this topic
- Changing only the sign of the gradient rather than taking the negative reciprocal.
- Assuming parallel lines must have different gradients.
- Substituting an endpoint instead of the midpoint for a perpendicular bisector.
- Reading a gradient before rearranging into y = mx + c form.
- Judging perpendicularity from a sketch instead of from the gradients.
Exam tips
- Always check a perpendicular gradient by multiplying the two values. The product must be exactly minus 1.
- Write the gradient as a fraction before inverting it. Whole numbers are easier to flip when written over 1.
- For a perpendicular bisector, do the midpoint first and the gradient second, then combine.
- Rearrange any equation into y = mx + c before reading anything from it.
- Substitute the given point back into your final equation to confirm it lies on the line.
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Parallel and Perpendicular Lines FAQs
What is the rule for perpendicular gradients?
The product of the gradients of two perpendicular lines is minus 1, so each gradient is the negative reciprocal of the other. To find it, write the gradient as a fraction, turn it upside down, and change the sign. Multiplying the two values back together should give exactly minus 1.
How do I find a line parallel to another through a given point?
Use the same gradient as the original line, since parallel lines have equal gradients. Write y equals that gradient times x plus c, substitute the coordinates of the given point, and solve for c to complete the equation.
What is a perpendicular bisector?
A line that crosses a line segment at its midpoint and at right angles to it. Find the midpoint of the segment, calculate the gradient of the segment, take the negative reciprocal of that gradient, then use the midpoint and the new gradient to form the equation.
Why is minus 2 not the perpendicular gradient of a line with gradient 2?
Changing the sign alone is not sufficient, because 2 multiplied by minus 2 gives minus 4 rather than minus 1. The gradient must also be inverted, giving minus one half, and 2 multiplied by minus one half does equal minus 1.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.