These twelve questions practise every skill in coordinate geometry: gradients, midpoints, lengths, line equations, parallel and perpendicular lines, and collinear points. They are ordered from easier to harder. All of them are original.
Work each question on paper, with a quick sketch where it helps. Then open the answer and compare your method, not only your final value. Use a calculator only where a question says so.
Questions and worked answers
Q1. Find the gradient of the line through (2, 1) and (6, 9).
Show answer
Rise = 9 − 1 = 8. Run = 6 − 2 = 4. Gradient = 8/4 = 2.
Check: going across 4 and up 8 matches 2 up for every 1 across. ✓
Q2. Find the gradient of the line through (−3, 7) and (1, −5).
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Rise = −5 − 7 = −12. Run = 1 − (−3) = 4. Gradient = −12/4 = −3.
Check: swapping the points gives 12/(−4) = −3. ✓
Q3. Find the midpoint of the points (−4, 6) and (10, −2).
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x = (−4 + 10)/2 = 6/2 = 3. y = (6 + (−2))/2 = 4/2 = 2. The midpoint is (3, 2).
Check: (3, 2) is 7 across from each end in x (−4 to 3 and 3 to 10) and 4 away in y from each end (6 to 2 and 2 to −2). ✓
Q4. Find the length of the segment joining (1, 2) and (5, 5).
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Differences: 5 − 1 = 4 and 5 − 2 = 3. Length = √(16 + 9) = √25 = 5.
Check: this is a 3-4-5 right-angled triangle. ✓
Q5. Find the exact length of the segment joining (−1, 3) and (4, −2).
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Differences: 4 − (−1) = 5 and −2 − 3 = −5. Length = √(25 + 25) = √50 = √(25 × 2) = 5√2.
Check: 5√2 ≈ 5 × 1.414 = 7.07, and √50 ≈ 7.07. ✓
Q6. Find the equation of the line with gradient 2 through (4, −3).
Show answer
y − (−3) = 2(x − 4), so y + 3 = 2x − 8, giving y = 2x − 11.
Check: at x = 4, y = 8 − 11 = −3. ✓
Q7. Find the equation of the line through (−2, 8) and (2, 2).
Show answer
Gradient = (2 − 8)/(2 − (−2)) = −6/4 = −3/2. Using (2, 2): y − 2 = −3/2(x − 2), so y − 2 = −3/2 x + 3, giving y = −3/2 x + 5.
Check with the other point: at x = −2, y = 3 + 5 = 8. ✓
Q8. Find the equation of the line parallel to 3x + y = 5 that passes through (1, 4).
Show answer
Rearrange: y = −3x + 5, so the gradient is −3. A parallel line has gradient −3. Then y − 4 = −3(x − 1), so y − 4 = −3x + 3, giving y = −3x + 7.
Check: at x = 1, y = −3 + 7 = 4. ✓
Q9. Find the equation of the line perpendicular to y = x/2 − 3 that passes through (2, −5).
Show answer
The gradient is 1/2. The perpendicular gradient is the negative reciprocal: −2. Then y − (−5) = −2(x − 2), so y + 5 = −2x + 4, giving y = −2x − 1.
Check: at x = 2, y = −4 − 1 = −5. ✓ Also (1/2) × (−2) = −1. ✓
Q10. Show that (2, −1), (5, 4) and (11, 14) are collinear.
Show answer
Gradient of the first two points: (4 − (−1))/(5 − 2) = 5/3. Gradient of the last two points: (14 − 4)/(11 − 5) = 10/6 = 5/3.
The gradients are equal and (5, 4) is common to both segments, so the three points are collinear.
Q11. M(−1, 4) is the midpoint of AB. A is (3, −2). Find the coordinates of B.
Show answer
x: 2 × (−1) − 3 = −5. y: 2 × 4 − (−2) = 10. So B is (−5, 10).
Check: the midpoint of (3, −2) and (−5, 10) is (−2/2, 8/2) = (−1, 4). ✓
Q12. P is (1, 1), Q is (5, 3) and R is (3, 7). Show that angle PQR is a right angle.
Show answer
Gradient of PQ = (3 − 1)/(5 − 1) = 2/4 = 1/2. Gradient of QR = (7 − 3)/(3 − 5) = 4/(−2) = −2.
The product is (1/2) × (−2) = −1, so PQ and QR are perpendicular. Because they meet at Q, angle PQR is a right angle.
Check with lengths: PQ² = 16 + 4 = 20, QR² = 4 + 16 = 20, PR² = 4 + 36 = 40, and 20 + 20 = 40. ✓
If you got these wrong
Use the type of error to decide where to go next.
- Q1, Q2 or Q10 wrong, usually a sign or an upside-down fraction: revisit calculating a gradient from two points and checking collinear points.
- Q3, Q4, Q5 or Q11 wrong, such as forgetting to halve or add the differences: revisit finding a midpoint and a segment length.
- Q6 or Q7 wrong, where the equation fails the point test: revisit writing the equation of a line through a point.
- Q8, Q9 or Q12 wrong, especially the flip-and-negate step: revisit identifying parallel and perpendicular lines.
Keep a short log of each wrong answer and the reason. The mistake log and retest queue can help you repeat a fresh question on the same skill a few days later. The non-calculator working trainer is handy for fraction steps.
If the same error returns after you have reviewed the lesson, a teacher can look at your actual working in online one-to-one Mathematics tuition.