NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Properties of quadrilaterals

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A craft design in Melbourne features a quadrilateral-shaped frame made from two congruent right-angled triangles joined along their hypotenuses to form a kite. The diagonals of the kite intersect at right angles, and the shorter diagonal is 58 cm long.

The longer diagonal bisects the shorter diagonal and is 87 cm long. What is the exact length, in centimetres, of one of the equal sides of the kite?

Express your answer as a simplified radical expression.

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Squares belong to two broader families. A square is both a rectangle and a rhombus. The arrows mean “is a type of”, not a step in a calculation.RectangleRhombusSquare
A square is both a rectangle and a rhombus. The arrows mean “is a type of”, not a step in a calculation.

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Worked solution

  1. The kite is formed by two congruent right-angled triangles joined along their hypotenuses. The diagonals intersect at right angles, and the shorter diagonal is 58 cm, while the longer diagonal is 87 cm.
  2. The longer diagonal (87 cm) bisects the shorter diagonal (58 cm), so each half of the shorter diagonal is 58 / 2 = 29 cm.
  3. The diagonals of a kite intersect at right angles, forming four right-angled triangles. Each half of the diagonals forms a leg of a right triangle, with the side of the kite as the hypotenuse.
  4. One right triangle has legs of 29 cm (half of shorter diagonal) and 43.5 cm (half of longer diagonal, since 87 / 2 = 43.5).
  5. Apply the Pythagorean theorem: c^2 = a^2 + b^2, where c is the hypotenuse (the equal side of the kite), a = 29 cm, and b = 43.5 cm.
  6. Substitute values: c^2 = 29^2 + 43.5^2.
  7. Calculate 29^2 = 841.
  8. Calculate 43.5^2 = (87/2)^2 = 7569 / 4 = 1892.25.
  9. Add: c^2 = 841 + 1892.25 = 2733.25.
  10. Take the square root: c = sqrt(2733.25). Convert 2733.25 to fraction: 2733.25 = 10933 / 4.
  11. Simplify sqrt(10933 / 4) = sqrt(10933) / 2. Factor 10933: 10933 = 10933 (check for square factors). 10933 ÷ 13 = 841, and 841 = 29^2. So 10933 = 13 × 29^2. Thus, sqrt(10933) = 29 × sqrt(13). Therefore, c = 29 × sqrt(13) / 2.
  12. The length of one of the equal sides of the kite is 29*sqrt(13)/2 cm. This is the simplified radical form.

Reference answer: 29*sqrt(13)/2 cm

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