Coordinates on a Line Segment
Section: Position and Transformation | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
Coordinate Recap
A point's position is given by coordinates (x, y): x is the horizontal position, y is the vertical position. A line segment is the straight path connecting two points.
Finding the Midpoint of a Line Segment
The midpoint is exactly halfway between two points - average the x-coordinates, and average the y-coordinates: M = ((x₁+x₂)/2, (y₁+y₂)/2).
The segment from A(2,3) to B(8,11): midpoint (5,7), and length 10 by Pythagoras
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Question: Find the midpoint of the line segment from A(2, 3) to B(8, 11).
- Step 1: Average the x-coordinates: (2 + 8) ÷ 2 = 5
- Step 2: Average the y-coordinates: (3 + 11) ÷ 2 = 7
- Answer: (5, 7)
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Question: Find the midpoint of the line segment from A(−4, 6) to B(2, −2).
- Step 1: Average the x-coordinates: (−4 + 2) ÷ 2 = −1
- Step 2: Average the y-coordinates: (6 + −2) ÷ 2 = 2
- Answer: (−1, 2)
Working Backwards from a Given Midpoint
If you're only given the midpoint, there isn't a single correct pair of endpoints - infinitely many pairs of A and B share the same midpoint. Any pair works as long as the x-coordinates average to the midpoint's x-coordinate, and the y-coordinates average to the midpoint's y-coordinate.
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Question: The line segment AB has a midpoint of (4, 6). Write possible coordinates for A and B.
- Step 1: The x-coordinates must sum to 2 × 4 = 8, and the y-coordinates must sum to 2 × 6 = 12
- Step 2: Choose any pair satisfying both totals, e.g. A = (0, 0) and B = (8, 12)
- Step 3 (check): ((0+8)/2, (0+12)/2) = (4, 6) ✓
- Answer: A = (0, 0), B = (8, 12) (one of many valid answers)
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Question: Using a quicker method, find a different valid pair of endpoints for the same midpoint (4, 6).
- Step 1: Pick any displacement (d, e) and move that far each way from the midpoint: A = (4 − d, 6 − e), B = (4 + d, 6 + e)
- Step 2: Using displacement (3, 1): A = (4 − 3, 6 − 1) = (1, 5), B = (4 + 3, 6 + 1) = (7, 7)
- Step 3 (check): ((1+7)/2, (5+7)/2) = (4, 6) ✓
- Answer: A = (1, 5), B = (7, 7) (also valid - any displacement works)
Finding a Point a Fraction Along a Line Segment
Find the horizontal and vertical differences between the two points, multiply each by the given fraction, then add the results to the starting point's coordinates.
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Question: Find the point 1/3 of the way from A(2, 3) to B(11, 15).
- Step 1: Differences: horizontal = 11 − 2 = 9, vertical = 15 − 3 = 12
- Step 2: Take 1/3 of each: 9 × 1/3 = 3, and 12 × 1/3 = 4
- Step 3: Add to A: (2 + 3, 3 + 4)
- Answer: (5, 7)
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Question: Find the point 3/4 of the way from A(0, 0) to B(8, 12).
- Step 1: Differences: horizontal = 8 − 0 = 8, vertical = 12 − 0 = 12
- Step 2: Take 3/4 of each: 8 × 3/4 = 6, and 12 × 3/4 = 9
- Step 3: Add to A: (0 + 6, 0 + 9)
- Answer: (6, 9)
Finding the Length of a Line Segment
The horizontal and vertical differences between two points form the two shorter sides of a right-angled triangle, with the line segment itself as the hypotenuse - so its length follows from Pythagoras' theorem: length = √((x₂−x₁)² + (y₂−y₁)²).
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Question: Find the length of the line segment from A(2, 3) to B(8, 11).
- Step 1: Differences: horizontal = 8 − 2 = 6, vertical = 11 − 3 = 8
- Step 2: length = √(6² + 8²) = √(36 + 64) = √100
- Answer: 10
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Question: Find the length of the line segment from A(−2, 1) to B(6, 7).
- Step 1: Differences: horizontal = 6 − (−2) = 8, vertical = 7 − 1 = 6
- Step 2: length = √(8² + 6²) = √(64 + 36) = √100
- Answer: 10
Real-World Applications
Points on a line segment are used in many contexts:
- GPS and mapping: finding a fair midpoint meeting location between two places.
- Construction: finding the centre point of a beam or wall.
- Computer graphics: interpolating points along a line for animation.
- Cartography: measuring straight-line distances between two points on a map.
- Sports: finding the halfway point on a field or track.
Exam Tips
Common Mistakes
MistakeAdding coordinates instead of averaging them when finding a midpoint
Fixthe midpoint formula divides the sum by 2 - always average, don't just add
MistakeApplying the fraction to only the x-coordinate (or only the y-coordinate)
Fixapply the fraction to BOTH the horizontal and vertical differences
MistakeMeasuring the fraction from the wrong end of the segment
Fixalways start from the point named first in the question
MistakeForgetting to take the square root at the end of a length calculation
Fixafter adding the squared differences, always take the square root to find the actual length
MistakeGetting confused by negative coordinates when subtracting
Fixwrite out the subtraction carefully, e.g. 6 − (−2) = 8, before squaring
MistakeAssuming there is only one correct pair of endpoints when working backwards from a midpoint
Fixinfinitely many pairs work - any A and B whose coordinates average to the given midpoint are valid
For Exams
- Learn the midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2).
- For a fraction along a segment: find the differences, multiply by the fraction, then add to the start point.
- Length uses Pythagoras: √((x₂−x₁)² + (y₂−y₁)²).
- Sketch the points on a quick grid if you're unsure - it helps catch mistakes.
- Check your answer makes sense: it should lie between A and B.
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