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

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A school in Brisbane is ordering custom quadrilateral-shaped desks for a new classroom. Each desk is designed as a kite with two pairs of adjacent equal sides.

The diagonals of the kite intersect at right angles, and one diagonal is 56 cm long while the other is 52 cm long. The longer diagonal bisects the shorter diagonal.

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 has diagonals intersecting at right angles, with one diagonal 56 cm and the other 52 cm. The longer diagonal (56 cm) bisects the shorter diagonal (52 cm), so it splits it into two equal parts of 26 cm each.
  2. The diagonals of a kite intersect at 90 degrees, forming four right-angled triangles. The two equal sides of the kite are the hypotenuses of two congruent right triangles formed by half of each diagonal.
  3. Each right triangle has legs of 26 cm (half of the shorter diagonal) and 28 cm (half of the longer diagonal, since 56 ÷ 2 = 28).
  4. To find the length of one of the equal sides of the kite, apply the Pythagorean theorem: c² = a² + b², where c is the hypotenuse, and a and b are the legs.
  5. Substitute the values: c² = 26² + 28².
  6. Calculate 26² = 676 and 28² = 784.
  7. Add: 676 + 784 = 1460.
  8. So c² = 1460. Therefore, c = sqrt(1460).
  9. Simplify sqrt(1460). Factor 1460: 1460 = 4 × 365 = 4 × 5 × 73. So sqrt(1460) = sqrt(4 × 365) = 2 × sqrt(365).
  10. 365 factors as 5 × 73, both prime, so no further simplification is possible.
  11. Thus, the length of one of the equal sides is 2 × sqrt(365) cm.
  12. Check: (2 × sqrt(365))² = 4 × 365 = 1460, which matches 26² + 28². The solution is correct.

Reference answer: 2*sqrt(365) cm

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