NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Three-dimensional coordinates

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Required skills (click to reveal)

In a community garden in Brisbane, a vertical trellis is positioned so that its base is at the origin (0, 0, 0) and its top point is at (79, 0, 22). A gardener places a small light at a point exactly halfway along the straight line from the base to the top of the trellis.

What is the exact z-coordinate of the light's position in metres?

Concept illustrations

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Three perpendicular coordinate directions. A schematic projection of perpendicular x,y,z axes. The point (2,3,6) is reached by changes 2 in x, 3 in y and 6 in z. Perspective spacing is illustrative, not an angle or distance measurement.xyz(2,3,6)236
A schematic projection of perpendicular x,y,z axes. The point (2,3,6) is reached by changes 2 in x, 3 in y and 6 in z. Perspective spacing is illustrative, not an angle or distance measurement.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The trellis extends from the base at (0, 0, 0) to the top at (79, 0, 22). The light is placed exactly halfway along the straight line connecting these two points.
  2. To find the midpoint of a line segment in 3D space, use the midpoint formula: ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2).
  3. Substitute the coordinates of the base (0, 0, 0) and the top (79, 0, 22) into the formula.
  4. Calculate the x-coordinate: (0 + 79)/2 = 79/2 = 39.5.
  5. Calculate the y-coordinate: (0 + 0)/2 = 0/2 = 0.
  6. Calculate the z-coordinate: (0 + 22)/2 = 22/2 = 11.
  7. The z-coordinate of the light is 11 metres.
  8. Verify the result: halfway between 0 and 22 is 11, which is correct.

Reference answer: 11

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