The Rope Course

Year 9 · Linear Relationships

# The Rope Course

Lines, lengths and crossings on a grid

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## Douglas's design

Two trees **20 m apart** become the lines x = −10 and x = 10. Two ropes join four points:

A(−10, 3) → B(10, 7)  ·  C(−10, 9) → D(10, 4)

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## How long is each rope?

A rope is the straight line between its two ends — the **longest side** of a right triangle (the *hypotenuse*):

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## Rope AB

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## Rope CD

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## The line of rope AB

Every straight line is — so find m, then c.

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## The line of rope CD

Same recipe — find m, then c for

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## Where do the ropes cross?

cross at about (3.3, 5.7) 👈

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## A third rope — perpendicular to AB

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## How high up the tree?

It reaches the left tree, where x = −10. Show the equation, **then** substitute:

55 m — far above the 9 m treetop! 😮

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## How long is the third rope?

From M(0, 5) to (−10, 55) — the distance formula again:

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## A fourth rope — let's fit it

From D(10, 4), tie it anywhere on the left tree from A(−10, 3) up to (−10, 6):

Slide the join point and watch **L** change — when is the rope shortest? longest?

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## Shortest and longest

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## The toolkit — all on one grid

lengths

each rope's line

intersection

perpendicular

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# Now go build it! 🌳

Four points, a grid, and a bit of algebra — that's the whole design.