Modelling problems on the coordinate plane

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Worksheet with answersDownload PDF

What this lesson covers

  • Two lines meet where their equations agree.
  • The shortest distance from a point to a line is along the perpendicular.
  • Finish with an answer in context, with units.
Key idea

The plan for a shortest distance

  • Find the perpendicular line through the point.
  • Find where it meets the original line.
  • Find the distance between those two points.
Worked example

Where two roads meet

  • Roads follow y = 2x + 1 and y = −x + 7.
  1. 2x + 1 = −x + 7set the y values equal
  2. 3x = 6, so x = 2
  3. y = 2(2) + 1 = 5
  4. They meet at (2, 5)
Worked example

The shortest pipe

  • A road follows y = 2x and a house is at H(10, 0), with units in kilometres.
  1. perpendicular gradient = −1/2
  2. y = −(1/2)(x − 10) = −(1/2)x + 5
  3. 2x = −(1/2)x + 5, so x = 2, y = 4
  4. d = √(8² + 4²) = √80
  5. The pipe is about 8.94 km

Common mistakes

  • Measuring straight across to the x-axis or y-axis.
    The shortest distance runs at right angles to the line.
  • Stopping at the meeting point.
    The question wants the distance, so finish with Pythagoras.

Your turn

  • 1. A road follows y = x and a house is at (6, 0). Where is the closest point on the road?
  • 2. How far is the house from the road?
Show answer

1. The perpendicular is y = −x + 6, meeting at (3, 3) 2. √18, about 4.24 units

Summary

What to take away

  • Turn the words into points and lines first.
  • Shortest distance means perpendicular.

More in AS 91256

Distance, midpoint and gradientNotesWriting the equation of a lineNotesParallel and perpendicular linesNotesModelling problems on the coordinate planeNotes

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