Perpendicular lines
Section: Coordinate geometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Finding the Gradient of a Perpendicular Line
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Two lines are perpendicular if they cross at a right angle. There is a simple relationship between the gradients of two perpendicular lines.
- If two lines are perpendicular, the product of their gradients is always -1: m_1 × m_2=-1
- To find the gradient of a line perpendicular to a given line, take the negative reciprocal of the given gradient: flip the fraction and change its sign
- A line with gradient 4 has a perpendicular gradient of -1/4; a line with gradient -2/3 has a perpendicular gradient of 3/2
Worked Example: Finding a Perpendicular Gradient
- Question: A line has gradient 2/5. Find the gradient of a line perpendicular to it.
- Step 1: Take the reciprocal of 2/5: 5/2
- Step 2: Change the sign: -5/2
- Answer: perpendicular gradient = -5/2
Common Mistakes
MistakeOnly flipping the fraction, or only changing the sign, but not doing both
Fixfinding a perpendicular gradient always requires two steps together - flip the fraction (reciprocal) AND change the sign
MistakeForgetting to write a whole number as a fraction first before taking its reciprocal, e.g. treating 3 as staying 3 instead of becoming 1/3
Fixrewrite a whole-number gradient as a fraction over 1 first (e.g. 3 = 3/1), then flip and negate as normal
Finding the Equation of a Perpendicular Line
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Once the perpendicular gradient is known, the equation of the perpendicular line can be found using the same substitution method as for any other line.
- First find the gradient of the given line, then take its negative reciprocal to get the perpendicular gradient
- Substitute the perpendicular gradient and the coordinates of the required point into y = mx + c, then solve for c
- The two perpendicular lines cross at exactly one point, at a right angle
Worked Example: Finding the Equation of a Perpendicular Line
- Question: Line AB passes through A(-2, 1) and B(6, 5). Find the equation of the line perpendicular to AB that passes through the point (2, 7).
- Step 1: Gradient of AB: (5-1/6-(-2))=(4/8)=(1/2)
- Step 2: Perpendicular gradient: -1÷(1/2)=-2
- Step 3: Substitute (2, 7) into y=-2x+c: 7=-2(2)+c 7=-4+c c=11
- Answer: y = -2x + 11
Common Mistakes
MistakeUsing the gradient of AB itself in the final equation, instead of switching to the perpendicular gradient
Fixthe final equation always uses the negative reciprocal gradient, not the original line's gradient
MistakeSubstituting the given point into the equation of AB, rather than the new perpendicular line, when solving for c
Fixthe point given belongs to the new perpendicular line - substitute it together with the perpendicular gradient
Perpendicular Horizontal and Vertical Lines
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Horizontal and vertical lines are a special case: a line perpendicular to a horizontal line is always vertical, and a line perpendicular to a vertical line is always horizontal.
- The negative reciprocal rule does not work directly for a horizontal line, since its gradient is 0 (there is no reciprocal of 0)
- A line perpendicular to a horizontal line y = c is always a vertical line, of the form x = k
- A line perpendicular to a vertical line x = c is always a horizontal line, of the form y = k
Worked Example: Finding a Line Perpendicular to a Horizontal Line
- Question: Line M has equation y = -3. Find the equation of the line perpendicular to M that passes through the point (5, 2).
- Step 1: Line M is horizontal (y = -3), so the perpendicular line must be vertical
- Step 2: A vertical line takes the form x = k, using the x-coordinate of the given point
- Answer: x = 5
Common Mistakes
MistakeTrying to apply the negative reciprocal rule to a horizontal or vertical line, leading to division by zero
Fixhorizontal and vertical lines are a special case - recognise them by inspection rather than using the formula
MistakeMixing up which direction the perpendicular line should be, e.g. drawing a horizontal line perpendicular to a horizontal line
Fixperpendicular to horizontal is always vertical, and perpendicular to vertical is always horizontal - the two directions swap
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