Strand 2 — Geometry & Trigonometry
Co-ordinate Geometry of the Line
1st Year · 2nd Year · 3rd Year (Junior Cert)
- ✓By the end of this lesson students will be able to calculate the distance between two points on the Cartesian plane.
- ✓By the end of this lesson students will be able to find the midpoint of a line segment.
- ✓By the end of this lesson students will be able to determine the slope (gradient) of a line given two points or its equation.
- ✓By the end of this lesson students will be able to find the equation of a line using various methods (e.g., point-slope form, two points).
- ✓By the end of this lesson students will be able to identify and apply the conditions for parallel and perpendicular lines.
Key concepts
The Cartesian plane is a two-dimensional plane defined by two perpendicular axes, the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0,0). Points on this plane are represented by ordered pairs (x, y), where x is the x-coordinate and y is the y-coordinate.
The distance between two points (x₁, y₁) and (x₂, y₂) on the Cartesian plane can be found using a formula derived from Pythagoras' theorem. It calculates the length of the line segment connecting the two points.
The midpoint of a line segment is the point that lies exactly halfway between its two endpoints. It is found by averaging the x-coordinates and averaging the y-coordinates of the two points.
The slope, often denoted by 'm', is a measure of the steepness of a line. It describes how much the y-coordinate changes for a given change in the x-coordinate. It is also referred to as 'rise over run'.
The general form of the equation of a straight line is ax + by + c = 0, where a, b, and c are constants, and a and b are not both zero. This form is often required for final answers in exams.
The slope-intercept form of the equation of a straight line is y = mx + c, where 'm' is the slope of the line and 'c' is the y-intercept (the point where the line crosses the y-axis). This form is useful for quickly identifying the slope and y-intercept.
The point-slope form is used to find the equation of a line when you know its slope 'm' and at least one point (x₁, y₁) that lies on the line.
Two distinct lines are parallel if and only if they have the exact same slope. Parallel lines never intersect.
Two lines are perpendicular if and only if the product of their slopes is -1. This means one slope is the negative reciprocal of the other. This condition applies unless one line is horizontal (slope 0) and the other is vertical (undefined slope).
Key facts to remember
- 1The distance between two points (x₁, y₁) and (x₂, y₂) is d = √((x₂ - x₁)² + (y₂ - y₁)²).
- 2The midpoint of the line segment joining (x₁, y₁) and (x₂, y₂) is M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
- 3The slope of a line passing through (x₁, y₁) and (x₂, y₂) is m = (y₂ - y₁)/(x₂ - x₁).
- 4The equation of a line can be written in general form ax + by + c = 0 or slope-intercept form y = mx + c.
- 5The point-slope form y - y₁ = m(x - x₁) is used to find the equation of a line given a point (x₁, y₁) and its slope m.
- 6Parallel lines have the same slope (m₁ = m₂).
- 7Perpendicular lines have slopes whose product is -1 (m₁ * m₂ = -1), unless one is horizontal (m=0) and the other is vertical (undefined slope).
- 8A horizontal line has a slope of 0 and an equation of the form y = k. A vertical line has an undefined slope and an equation of the form x = k.
Worked examples
Example 1
Find the distance between the points A(2, -3) and B(-4, 5). Also, find the midpoint of the line segment AB.
Answer
The distance between A and B is 10 units. The midpoint of AB is (-1, 1).
Remember to be careful with negative signs, especially when subtracting a negative number.
Example 2
Find the slope of the line passing through the points P(1, 7) and Q(4, 1). Then, find the equation of this line in the form ax + by + c = 0.
Answer
The slope of the line is -2. The equation of the line is 2x + y - 9 = 0.
You could also use point Q(4, 1) to find the equation; the final result will be the same.
Example 3
A line L passes through the point (3, -2) and is parallel to the line 2x + 3y - 6 = 0. Find the equation of line L. Then, find the equation of a line M that passes through (3, -2) and is perpendicular to the line 2x + 3y - 6 = 0.
Answer
The equation of line L (parallel) is 2x + 3y = 0. The equation of line M (perpendicular) is 3x - 2y - 13 = 0.
Always convert the given line's equation to y = mx + c form to easily find its slope.
Common mistakes
- ✗**Sign Errors**: Incorrectly handling negative numbers when substituting into formulas, especially in the distance and slope formulas.
- ✗**Incorrect Formula Substitution**: Mixing up x and y coordinates or using the wrong formula for distance, midpoint, or slope.
- ✗**Algebraic Errors**: Mistakes when rearranging equations, particularly when converting to the ax + by + c = 0 form or clearing fractions.
- ✗**Confusing Parallel and Perpendicular Conditions**: Forgetting to use the negative reciprocal for perpendicular slopes, or incorrectly applying the parallel condition.
- ✗**Not Simplifying Answers**: Leaving square roots unsimplified or not presenting the final equation of a line in the requested form (e.g., ax + by + c = 0).
Exam tips
- ★**Label Your Points**: Always label your coordinates as (x₁, y₁) and (x₂, y₂) at the start of a problem to avoid substitution errors.
- ★**Write Down the Formula**: Before substituting values, write down the relevant formula. This helps you remember it and can earn partial credit even if you make a calculation error.
- ★**Check Your Work**: After calculating, do a quick visual check. Does the slope make sense (positive/negative)? Does the midpoint appear to be in the middle of the segment?
- ★**Be Careful with Negatives**: Use brackets when substituting negative numbers into formulas to prevent common sign errors, e.g., (5 - (-3)) should be written as (5 - (-3)) or (5+3).
- ★**Practise Rearranging Equations**: Be proficient in converting between the y = mx + c form and the ax + by + c = 0 form, as questions often require the answer in a specific format.
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