Real-World Geometry Applications

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Date:

🏗️ Part 1: Geometry in Architecture and Engineering

1. Look at the Auckland Sky Tower image. Which geometric shapes can you identify in its design?

Circles

Triangles

Rectangles

Cylinders

2. A bridge cable forms a chord in a circular arch. If the radius of the arch is 15 metres and the chord is 24 metres long, what is the diameter of the arch?

15 metres

24 metres

30 metres

39 metres

3. In a building's blueprint, two parallel support beams are crossed by a diagonal brace. If one angle measures 65°, what is the measure of the alternate interior angle?

4. Extension: A triangular roof truss has sides of 8m, 6m, and 10m. Use Pythagoras' theorem to determine if this forms a right-angled triangle. Show your working.

🎨 Part 2: Geometry in Art and Cultural Design

5. Māori tukutuku panels often feature rotational symmetry. What type of transformation creates a pattern that looks the same after being rotated 90°?

6. A tessellation pattern uses regular hexagons. How many degrees is each interior angle of a regular hexagon?

108°

120°

135°

144°

7. Describe one example of reflection symmetry you might find in traditional New Zealand architecture or art:

📐 Part 3: Scale Factors in Real-World Applications

8. An architect creates a model house at a scale of 1:50. If the actual house is 12 metres tall, how tall is the model?

Model height: _________ centimetres

9. A rectangular garden bed measures 4m × 3m. If it's scaled up by a factor of 2, what happens to its area?

Area doubles (becomes 2 times larger)

Area triples (becomes 3 times larger)

Area quadruples (becomes 4 times larger)

Area stays the same

10. Challenge Problem: A cylindrical water tank has a radius of 2m and height of 5m. If a scale model is built at 1:10 scale, calculate both the model's dimensions and how the volume compares to the original.

11. Explain why understanding scale factors is important for architects and engineers: