John made a triangular poster for the coming mathematics competition. It was in the shape of an equilateral triangle with side 30 cm as shown in the diagram below. [diagram of an equilateral triangle with each side labelled 30 cm] His teacher liked the poster very much and asked him to make another similar one but with an area twice that of the first poster. What should be the length of each side of the new poster? Show your working clearly. Answer

1. Identify the area scale factor of the new poster compared with the original poster.

The new poster is similar to the first poster and has twice the area.

So the area scale factor is
$2.$

2. Find the corresponding length scale factor from the area scale factor.

For similar figures,
$\text{area scale factor}=(\text{length scale factor})^2.$

Hence
$(\text{length scale factor})^2=2$
$\text{length scale factor}=\sqrt{2}.$

3. Apply the length scale factor to the original side length.

Original side length $=30\text{ cm}$.

New side length
$=30\times \sqrt{2}$
$=30\sqrt{2}\text{ cm}$
$\approx 42.4\text{ cm}.$