NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Gradient, midpoint and distance

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In a Cartesian plane used for planning a new school materials order in Sydney, two delivery points are located at A(21, 42) and B(42, 21). Find the exact gradient of the line segment AB.

Express your answer as a simplified fraction or integer.

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Rise, run and midpoint. From (1,2) to (5,6), run = 4 and rise = 4. Gradient = 1, midpoint = (3,4) and distance = √32 = 4√2. Equal coordinate scales.(1,2)(5,6)rise 4run 4(3,4)
From (1,2) to (5,6), run = 4 and rise = 4. Gradient = 1, midpoint = (3,4) and distance = √32 = 4√2. Equal coordinate scales.

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Worked solution and answer (click to reveal)

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Worked solution

  1. Identify the coordinates of points A and B: A(21, 42) and B(42, 21).
  2. The gradient of a line segment between two points (x1, y1) and (x2, y2) is given by the formula: (y2 - y1) / (x2 - x1).
  3. Substitute the coordinates into the gradient formula: (21 - 42) / (42 - 21).
  4. Calculate the difference in y-coordinates: 21 - 42 = -21.
  5. Calculate the difference in x-coordinates: 42 - 21 = 21.
  6. Form the fraction: -21 / 21.
  7. Simplify the fraction: -21 ÷ 21 = -1.
  8. The gradient is a simplified integer, so no further reduction is needed.
  9. Verify the result by checking that the same gradient is obtained regardless of point order: (42 - 21) / (21 - 42) = 21 / (-21) = -1.
  10. The gradient is constant along the line, confirming the result is correct.
  11. The gradient of line segment AB is -1.
  12. Final answer: -1.

Reference answer: -1

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