A car workshop hangs a kite-shaped promotional banner above its entrance to attract passing customers on the main road. The diagonals of the kite are 90 cen...
Question 1 Report
A car workshop hangs a kite-shaped promotional banner above its entrance to attract passing customers on the main road. The diagonals of the kite are 90 centimetres and 50 centimetres, and they cross each other at right angles above the doorway.
- Work out the area of the banner. (3)
- The workshop orders a second banner that is mathematically similar but with diagonals twice as long. Work out the area of the second banner. (2)
Answer Details
The banner is a kite with perpendicular diagonals, so its area is \( \dfrac{1}{2} \times d_1 \times d_2 \). For a similar shape with every diagonal doubled, area scales by the square of that linear scale factor.
- With diagonals 90 cm and 50 cm:\[ \text{Area} = \frac{1}{2} \times 90 \times 50 = 2250 \text{ cm}^2 \][3] (1 for \(\frac{1}{2} \times 90\), 1 for multiplying by 50, 1 for 2250 cm\(^2\))
- The second banner is similar with diagonals twice as long, so its linear scale factor is 2 and its area scale factor is \(2^2 = 4\):\[ \text{New area} = 2250 \times 4 = 9000 \text{ cm}^2 \][2] (1 for the area scale factor 4, 1 for 9000 cm\(^2\))
Doubling every length of a similar shape quadruples its area rather than doubling it, because area scales with the square of the linear scale factor; this is worth spotting quickly rather than recalculating the area from scratch with doubled diagonals.
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