Two triangular flags used at a music festival are mathematically similar, shown in the diagram. Flag A has an area of \(72\) cm\(^2\) and a base of \(12\) c...
Question 1 Report
Two triangular flags used at a music festival are mathematically similar, shown in the diagram. Flag A has an area of \(72\) cm\(^2\) and a base of \(12\) cm. Flag B is an enlargement of Flag A with an area of \(162\) cm\(^2\).
- Work out the height of Flag A. (2)
- Find the area scale factor from Flag A to Flag B, and hence the length scale factor. (2)
- Work out the base length of Flag B. (1)
Answer Details
For similar triangles, the standard area formula \(\text{area}=\frac{1}{2}\times\text{base}\times\text{height}\) finds an unknown height, and the ratio of areas of similar shapes equals the square of the ratio of their corresponding lengths.
(a) Substituting the known area and base of Flag A into the area formula:
\[72=\dfrac{1}{2}\times12\times h\]so \(72=6h\), giving \(h=12\) cm. [2 marks]
(b) The area scale factor from Flag A to Flag B is the ratio of their areas:
\[\dfrac{162}{72}=2.25\]Since area scales with the square of the length scale factor, the length scale factor is the square root of this:
\[\sqrt{2.25}=1.5\] [2 marks](c) Multiplying Flag A's base by the length scale factor:
\[12\times1.5=18 \text{ cm}\] [1 mark]Finding the area scale factor first and only then taking its square root is essential; applying \(2.25\) directly to a length (rather than \(1.5\)) would badly overstate Flag B's base.
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