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Cambridge IGCSE0607

Length and midpoint

International Mathematics 0607 Chapter Notes

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Length and midpoint
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In this chapter

  1. 1Midpoint of a line segment
  2. 2Midpoints with negative and decimal coordinates
  3. 3Length of a line segment
  4. 4Length as a surd
  5. 5Midpoint and length in one question

1. Midpoint of a line segment

The midpoint of a line segment is the point halfway between its end points.

To find its coordinates, add the two x-coordinates and halve. Then add the two y-coordinates and halve.

Give the answer as a pair of coordinates, (x, y).

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Examiner insight

“A few candidates found the gradient of AB rather than the mid-point.”

Worked example 12 marks

Line L has equation y = 2x - 5. Line L crosses the y-axis at point A, so A = (0, -5). Point B has coordinates (4, 3).

(d) Find the coordinates of the midpoint of AB.

  1. 1

    x = (0 + 4)/2 = 2 [1]

  2. 2

    y = (-5 + 3)/2 = -1, so the midpoint is (2, -1) [1]

2. Midpoints with negative and decimal coordinates

Add negative numbers with care: -6 + 3 = -3. Halve the sum afterwards, so -3 ÷ 2 = -1.5.

A midpoint can have decimal coordinates. Do not round them to whole numbers.

A sketch shows roughly where the midpoint lies. Use it to check, but work from the coordinates.

Examiner insight

“Here some candidates seemed to think that the midpoint must be a vertex on the grid so their answers only used whole numbers.” “A diagram is a good idea to see roughly where the mid-point will be, but should not be used instead of working with the individual co-ordinates.”

Worked example 12 marks

Do not use a calculator.

A is the point (-6, 2) and B is the point (3, -5). Find the coordinates of the midpoint of AB.

  1. 1

    x = (-6 + 3)/2 = (-3)/2 = -1.5 [1]

  2. 2

    y = (2 + (-5))/2 = (-3)/2 = -1.5, so the midpoint is (-1.5, -1.5) [1]

3. Length of a line segment

To calculate the length of a line segment, find the difference in the x-coordinates and the difference in the y-coordinates. These are the two shorter sides of a right-angled triangle.

Square each difference, add the squares, then take the square root. The order of subtraction does not matter, because squares are never negative.

AB = √((x₂ - x₁)² + (y₂ - y₁)²)

Examiner insight

“The subtraction of the coordinates sometimes had an arithmetical error for example 1 – (–5) was sometimes given as 4.” “Some candidates subtracted the squares instead of adding.”

Worked example 13 marks

A is the point (-5, 8) and B is the point (1, -1).

(a) Calculate the length of AB.

  1. 1

    differences: 1 - (-5) = 6 and -1 - 8 = -9 [1]

  2. 2

    AB² = 6² + (-9)² = 36 + 81 = 117 [1]

  3. 3

    AB = √(117) = 10.8 to 3 significant figures. On the calculator, enter √((6² + 9²)) and read 10.8167 [1]

4. Length as a surd

On a non-calculator paper, a question may ask for the length as a surd. Leave the square root in the answer. Do not use a decimal.

For AB² = 160, a mark scheme credits 4√10 or √(160) as the final answer. The two are equal: √(160) = √(16 × 10) = 4√10.

When a question asks for the form √p, stop at the square root, as in √80.

Examiner insight

“Several did not give the answer in surd form as required.”

Worked example 13 marks

A is the point (-4, 3) and B is the point (8, -1).

(a) Find the length of AB. Give your answer as a surd.

  1. 1

    differences: 8 - (-4) = 12 and -1 - 3 = -4 [1]

  2. 2

    AB² = 12² + (-4)² = 144 + 16 = 160 [1]

  3. 3

    AB = √(160) = √(16 × 10) = 4√10 [1]

5. Midpoint and length in one question

A question may ask for the midpoint and a length. Do them as two separate pieces of working.

The midpoint is halfway along the segment. The distance from an end point to the midpoint is therefore half the length of the whole segment.

Worked example 14 marks

Do not use a calculator.

A is the point (1, -2) and B is the point (7, 6). M is the midpoint of AB.

(a) Find the coordinates of M.

(b) Find the length of AM.

  1. 1

    (a) x = (1 + 7)/2 = 4 and y = (-2 + 6)/2 = 2, so M = (4, 2) [1]

  2. 2

    (b) differences from A to M: 4 - 1 = 3 and 2 - (-2) = 4 [1]

  3. 3

    AM² = 3² + 4² = 9 + 16 = 25 [1]

  4. 4

    AM = √25 = 5 [1]

Recap

  • Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2): add, then halve.
  • A midpoint can have decimal coordinates. Do not round them.
  • Length = √((x₂ - x₁)² + (y₂ - y₁)²): subtract, square, add, square root.
  • Take care with signs: 1 - (-5) = 6, not 4.
  • Never measure a length the question asks you to calculate.
  • With a calculator, give a length correct to 3 significant figures.
  • Without a calculator, leave the length as a surd, such as √(160) or 4√10.

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 segment from (2, 6) to (8, 10).2 marks
  2. Do not use a calculator. Find the midpoint of the line segment from (-4, 7) to (6, -3).2 marks
  3. Find the midpoint of the line segment from (-5, -2) to (2, 3).2 marks
  4. Do not use a calculator. Calculate the length of the line segment from (1, 2) to (4, 6).3 marks
  5. Do not use a calculator. Calculate the length of the line segment from (-3, 1) to (5, -5).3 marks
  6. Calculate the length of the line segment from (-2, 4) to (7, -3). Give your answer correct to 3 significant figures.3 marks
  7. Do not use a calculator. The points P and Q have coordinates (2, -1) and (-4, 5). Find the length of PQ. Give your answer as a simplified surd.3 marks
  8. Do not use a calculator. A is the point (-3, -1) and B is the point (5, 5). M is the midpoint of AB. Find the coordinates of M and the length of AM.4 marks
  9. A park is drawn on a coordinate grid. 1 unit on each axis represents 200 m. The gate G is at (-4, -1), the fountain F is at (2, 7) and the cafe C is at (8, -3). A bench M is at the midpoint of the straight path FC. (a) Find the coordinates of M. (b) Sana walks in a straight line from G to M, then in a straight line from M to C. Calculate the total distance she walks. Give your answer in km correct to 3 significant figures.6 marks
  10. Do not use a calculator. A triangular sail is drawn on a coordinate grid with corners A(-1, 4), B(5, 2) and C(1, -2). A straight seam joins A to M, the midpoint of BC. (a) Show that the sides AB and AC have the same length. (b) Find the coordinates of M. (c) Show that the seam AM has the same length as the side BC. Give this length as a simplified surd.7 marks

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