NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Area and perimeter of composite shapes

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A library in Perth, Western Australia, is constructing a new reading nook shaped as a composite figure. The base is a rectangle measuring 93 metres in length and 11 metres in width.

Attached to one of the 93-metre sides is a right-angled triangular extension, with the right angle at the corner of the rectangle, and the two legs of the triangle measuring 93 metres and 11 metres respectively. The hypotenuse of the triangle is not part of the outer boundary.

Calculate the exact total area of the composite shape in square metres. Express your answer as an exact fraction or simplified radical expression.

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.

Subtract a rectangular corner cut-out. The outer rectangle is 10 cm by 6 cm. Removing a 2 cm by 3 cm top-right corner leaves 60 − 6 = 54 cm². All dimensions are supplied; drawing is to one consistent scale.10 cm6 cm2 cm3 cm54 cm²
The outer rectangle is 10 cm by 6 cm. Removing a 2 cm by 3 cm top-right corner leaves 60 − 6 = 54 cm². All dimensions are supplied; drawing is to one consistent scale.
Area and perimeter of a rectangle. 6 × 4 = 24 square units inside; 6 + 4 + 6 + 4 = 20 units around.6 units across4
6 × 4 = 24 square units inside; 6 + 4 + 6 + 4 = 20 units around.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The composite shape consists of a rectangle and a right-angled triangle attached to one of its longer sides. The rectangle has length 93 m and width 11 m.
  2. The triangle has legs of 93 m and 11 m, with the right angle at the corner of the rectangle. This means the triangle shares the 93 m side with the rectangle.
  3. The area of a rectangle is given by length × width. Substitute the values: 93 m × 11 m.
  4. Calculate: 93 × 11 = 93 × (10 + 1) = 93×10 + 93×1 = 930 + 93 = 1023. So, area of rectangle = 1023 m².
  5. The area of a right-angled triangle is (1/2) × base × height. Here, base = 93 m, height = 11 m.
  6. Substitute: (1/2) × 93 × 11. First compute 93 × 11 = 1023, then multiply by 1/2.
  7. 1023 × 1/2 = 1023/2 = 511.5. So, area of triangle = 511.5 m².
  8. Total area = area of rectangle + area of triangle = 1023 + 511.5.
  9. 1023 + 511.5 = 1534.5. So, total area = 1534.5 m².
  10. Express 1534.5 as a fraction: 1534.5 = 15345/10 = 3069/2.
  11. The exact area is 3069/2 square metres. This is a simplified fraction and cannot be reduced further.
  12. The total area of the composite shape is exactly 3069/2 m².

Reference answer: 3069/2 m²

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