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Cambridge IGCSE0580

Length and midpoint

Mathematics 0580 Chapter Notes

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Length and midpoint
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In this chapter

  1. 1Length of a line segment
  2. 2Exact answers and accuracy
  3. 3Midpoint of a line segment

1. Length of a line segment

A line segment joins two points. To find its length, make a right-angled triangle on it. The horizontal side is the difference of the x-coordinates. The vertical side is the difference of the y-coordinates.

Then use Pythagoras' theorem: the length is the hypotenuse.

Subtract carefully. If a coordinate is negative, the subtraction has a double negative, such as 5 - (-3) = 8. Add the two squares, then take the square root.

length = √((x₂ - x₁)² + (y₂ - y₁)²)

Examiner insight

“Errors in the method included omitting the squaring steps, incorrect subtraction of the –5, or multiplication rather than addition of the two squared parts of the calculation.”

Worked example 13 marks

A is the point (5, 2) and B is the point (-3, 6). Calculate the length of AB.

  1. 1

    differences: 5 - (-3) = 8 and 2 - 6 = -4 [1]

  2. 2

    AB² = 8² + (-4)² = 64 + 16 = 80 [1]

  3. 3

    AB = √80 = 8.944… = 8.94 [1]

2. Exact answers and accuracy

Without a calculator, keep the working exact. Leave a root that is not a whole number as a square root.

With a calculator, give an answer correct to 3 significant figures unless the question says otherwise. Write down the more accurate value from your calculator first, then round.

An answer rounded too far, such as 8.9 for 8.944…, loses the accuracy mark. The working still earns the method marks.

Examiner insight

“Some candidates did not give sufficient accuracy in their answers, instead rounding to 8.9 without a more accurate value being seen; usually they gained 2 marks for the method shown.” “In addition the third mark was lost by candidates who gave the length as 18.9 instead of 19.0 correct to 3 significant figures.”

Worked example 13 marks

Do not use a calculator. C is the point (-2, 1) and D is the point (4, 9). Find the length of CD.

  1. 1

    differences: 4 - (-2) = 6 and 9 - 1 = 8 [1]

  2. 2

    CD² = 6² + 8² = 36 + 64 = 100 [1]

  3. 3

    CD = √(100) = 10 [1]

3. Midpoint of a line segment

The midpoint is halfway along the segment. Its coordinates are the average of the x-coordinates and the average of the y-coordinates.

Add the two x-values and halve. Do the same for the y-values. Never subtract.

The answer is a pair of coordinates, so write it as a point: (x, y).

midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Examiner insight

“Nearly all candidates realised the midpoint was given by the average of the x– and y– coordinates of the points A and B.” “An error in finding the midpoint was to subtract the co-ordinates and divide by 2, resulting in (3, 7).”

Worked example 12 marks

P is the point (0, 4) and Q is the point (1, 8). Find the coordinates of the midpoint of PQ.

  1. 1

    x: (0 + 1)/2 = 0.5 [1]

  2. 2

    y: (4 + 8)/2 = 6, so the midpoint is (0.5, 6) [1]

Recap

  • Length of a segment: Pythagoras' theorem on the x and y differences.
  • Add the squares, then take the square root.
  • Take care with negative coordinates: 5 - (-3) = 8.
  • Show a more accurate value before rounding to 3 significant figures.
  • Midpoint: the average of the x-values and the average of the y-values.
  • Add the coordinates for a midpoint. Subtract them for a length.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Find the coordinates of the midpoint of the line joining (2, 6) and (8, 10).2 marks
  2. Do not use a calculator. Find the coordinates of the midpoint of the line joining (-4, 3) and (6, -9).2 marks
  3. Do not use a calculator. M is the midpoint of AB. A is the point (1, -4) and M is the point (3, 2). Find the coordinates of B.3 marks
  4. Do not use a calculator. Find the length of the line joining (1, 2) and (4, 6).3 marks
  5. Do not use a calculator. P is the point (-3, -1) and Q is the point (9, 4). Find the length of PQ.3 marks
  6. Use a calculator. M is the point (-2, 7) and N is the point (5, -3). Calculate the length of MN. Give your answer correct to 3 significant figures.3 marks
  7. Use a calculator. R is the point (-6, 2) and S is the point (3, -5). Calculate the length of RS. Give your answer correct to 3 significant figures.3 marks
  8. Do not use a calculator. E is the point (1, -2) and F is the point (3, 2). Find the length of EF. Leave your answer as a square root.3 marks
  9. Do not use a calculator. A straight bridge runs from A to B on a grid where each unit is 1 m. A is the point (-4, 7). A support stands at the midpoint of the bridge, at M(2, -1). Find the coordinates of B, and the length of the bridge.5 marks
  10. Use a calculator. On a farm map, each unit represents 250 m. A well is at W(-7, 4) and a gate is at G(5, -2). A pump is at P, the midpoint of WG. A tank is at T(3, 9). A pipe runs in a straight line from W to P, then in a straight line from P to T. Calculate the total length of the pipe. Give your answer in kilometres, correct to 3 significant figures.6 marks

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