NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Classifying polygons

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In a geometry workshop at a library in Melbourne, students are classifying quadrilaterals using paper strips. One student constructs a quadrilateral with two pairs of adjacent equal sides and perpendicular diagonals.

The diagonals intersect at right angles, and one diagonal is 26 cm long and bisects the other diagonal. If the area of the quadrilateral is 44 cm², what is the length of the other diagonal?

Express your answer as an exact value.

Concept illustrations

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Triangulating a pentagon. Two diagonals from one vertex split this pentagon into three triangles. Its interior-angle sum is 3 × 180° = 540°.(5 − 2) × 180° = 540°
Two diagonals from one vertex split this pentagon into three triangles. Its interior-angle sum is 3 × 180° = 540°.
Adjacent angles on a straight line. The straight angle is 180°. The ray splits it into 65° and 115°, so the two adjacent angles sum to 180°. Drawing is schematic.65°115°65° + 115° = 180°
The straight angle is 180°. The ray splits it into 65° and 115°, so the two adjacent angles sum to 180°. Drawing is schematic.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The quadrilateral has two pairs of adjacent equal sides and perpendicular diagonals, which identifies it as a kite.
  2. One diagonal is 26 cm and bisects the other diagonal, meaning it is the axis of symmetry and the other diagonal is split into two equal parts.
  3. The area of a kite is given by the formula: Area = (d1 × d2) / 2, where d1 and d2 are the lengths of the diagonals.
  4. Substitute the known values: Area = 44 cm², d1 = 26 cm. Let d2 be the unknown diagonal length.
  5. Set up the equation: 44 = (26 × d2) / 2.
  6. Multiply both sides by 2: 88 = 26 × d2.
  7. Divide both sides by 26: d2 = 88 / 26.
  8. Simplify the fraction: 88 ÷ 2 = 44, 26 ÷ 2 = 13, so d2 = 44 / 13.
  9. The fraction 44/13 cannot be simplified further, so d2 = 44/13 cm.
  10. Check: (26 × 44/13) / 2 = (1144 / 13) / 2 = 88 / 2 = 44 cm². This matches the given area.
  11. The length of the other diagonal is 44/13 cm.
  12. The answer is expressed as an exact value, as required.

Reference answer: 44/13 cm

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