GCSE Further Maths · Topic 2
Coordinate geometry
Coordinate geometry turns shapes into equations.
We will find gradients, midpoints and distances, write equations of straight lines, and spot parallel and perpendicular lines.
Then we meet the equation of a circle and its tangents.
- 2 subtopics
- 84 questions
- Free: 2.1 notes and 10 questions
In this topic
- 2.1Straight lines: gradient, midpoint, distance, and parallel and perpendicular linesFree below · 33 questions
- 2.2Circles: the equation of a circle, completing the square and tangentsOpen in Full Marks · 51 questions
2.1 · Free notes
Straight lines: gradient, midpoint, distance, and parallel and perpendicular lines
Key points
- Gradient m = (y₂ − y₁)/(x₂ − x₁). Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2). Distance = √((x₂ − x₁)² + (y₂ − y₁)²).
- Equation of a line: y = mx + c, or through (x₁, y₁) with gradient m: y − y₁ = m(x − x₁). General form: ax + by + c = 0.
- Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to −1: m₂ = −1/m₁.
- To find where two lines meet, solve their equations simultaneously. To show points are collinear, show they lie on one line (equal gradients between pairs).
- The perpendicular bisector of AB passes through the midpoint of AB with gradient −1/(gradient of AB).
In more detail
- Worked example: find the perpendicular bisector of A(1, 3) and B(5, 7). Midpoint (3, 5); gradient of AB = 1; perpendicular gradient −1; y − 5 = −(x − 3), so y = −x + 8.
- Area of a triangle: if two sides are perpendicular, area = ½ × base × height using their lengths from the distance formula.
- Exam tip: rearrange into y = mx + c before reading off a gradient.
Practice
Straight lines: gradient, midpoint, distance, and parallel and perpendicular lines: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. What is the gradient of the line through (1, 2) and (4, 11)?
- 3
- 1/3
- 9
- −3
Show answer
A: 3. (11 − 2)/(4 − 1) = 3.
2. What is the midpoint of (−2, 5) and (6, 1)?
- (4, 6)
- (8, −4)
- (2, 3)
- (2, 2)
Show answer
C: (2, 3). ((−2 + 6)/2, (5 + 1)/2) = (2, 3).
3. What is the distance between (1, 1) and (4, 5)?
- 7
- 5
- 25
- √7
Show answer
B: 5. √(3² + 4²) = 5.
4. A line has gradient 2/3. What is the gradient of a perpendicular line?
- 3/2
- −2/3
- 2/3
- −3/2
Show answer
D: −3/2. m₂ = −1/m₁.
5. Find the equation of the line with gradient 4 through (2, 3).
- y = 4x − 5
- y = 4x + 3
- y = 4x − 2
- y = 3x + 4
Show answer
A: y = 4x − 5. y − 3 = 4(x − 2), so y = 4x − 5.
6. Which line is parallel to y = 3x − 7?
- y = −3x + 2
- y = x/3
- y = 3x + 2
- y = −x/3 + 1
Show answer
C: y = 3x + 2. Parallel lines have the same gradient, 3.
7. What is the gradient of 2x + 5y = 10?
- 2/5
- −2/5
- 5/2
- −5/2
Show answer
B: −2/5. 5y = −2x + 10, so y = −2x/5 + 2.
8. Where does y = 2x + 1 cross y = −x + 7?
- (3, 4)
- (1, 3)
- (6, 13)
- (2, 5)
Show answer
D: (2, 5). 2x + 1 = −x + 7 gives x = 2, y = 5.
9. The perpendicular bisector of A(0, 0) and B(4, 2) passes through which point?
- (2, 1)
- (4, 2)
- (0, 0)
- (1, 2)
Show answer
A: (2, 1). It passes through the midpoint (2, 1).
10. Which pair of gradients shows perpendicular lines?
- 2 and 1/2
- 2 and −2
- 2 and −1/2
- 3 and 3
Show answer
C: 2 and −1/2. 2 × (−1/2) = −1.
Keep going in Full Marks
Full Marks has 33 questions on Straight lines: gradient, midpoint, distance, and parallel and perpendicular lines (23 more than here), set out in practice levels. You also get notes and 3D models for all 2 subtopics of Coordinate geometry, plus games, timed exams and past papers.