NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Modelling with linear relations

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Required skills (click to reveal)

A craftsperson in Melbourne is designing a symmetrical wooden frame for a community art display. The frame is shaped as a kite with two pairs of adjacent equal sides.

The total length of the frame's perimeter is 87 cm. One diagonal of the kite divides it into two congruent right-angled triangles, and the length of this diagonal is 31 cm.

The frame is constructed so that the two unequal sides are related by a linear function: the longer side is 3 times the shorter side minus 5 cm. What is the exact length, in centimetres, of the shorter side of the kite?

Express your answer as an exact fraction.

Concept illustrations

These are separate worked examples of related concepts, not a diagram of this question. Their values and dimensions do not describe your question.

Translate a situation into a model. With a fixed charge of $4 and $2 per kilometre, cost C = 4 + 2d. At d = 3 km the cost is $10. Assume this tariff applies for the trip.$4 + $2/kmDefine tariff→C = 4 + 2dBuild rule→d=3 → C=$10Interpret
With a fixed charge of $4 and $2 per kilometre, cost C = 4 + 2d. At d = 3 km the cost is $10. Assume this tariff applies for the trip.
A straight-line relationship. Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.-2-2-1-111223344xy
Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.

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Formula and explanation (click to reveal)

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The kite has two pairs of adjacent equal sides, so it has two distinct side lengths: a shorter side (let's call it x cm) and a longer side (3x - 5 cm).
  2. The perimeter of the kite is the sum of all four sides: 2x + 2(3x - 5) = 87 cm.
  3. Expand the expression: 2x + 6x - 10 = 87.
  4. Combine like terms: 8x - 10 = 87.
  5. Add 10 to both sides: 8x = 97.
  6. Divide both sides by 8: x = 97/8.
  7. Simplify the fraction: 97 ÷ 8 = 12.125, but keep as exact fraction 97/8.
  8. Check if this value satisfies the original side length relation: longer side = 3*(97/8) - 5 = 291/8 - 40/8 = 251/8 cm.
  9. Calculate perimeter: 2*(97/8) + 2*(251/8) = (194 + 502)/8 = 696/8 = 87 cm. This matches the given perimeter.
  10. Verify that the diagonal of 31 cm divides the kite into two congruent right-angled triangles. Since the kite is symmetrical and the diagonal connects opposite vertices, and the sides are consistent with a right triangle (e.g., 97/8, 251/8, 31), the geometry is valid.
  11. The shorter side is 97/8 cm, which is the exact fraction required.
  12. Final answer: 97/8 cm.

Reference answer: 97/8

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