The area of a figure is the measure of the extent of a two dimensional figure which is usually measured as the square units of the dimensions of the figure.
For example, given a figure with dimensions, say k, the area of the figure is given by [tex]k^2[/tex].
Given that one
similar figure has an area that is nine times the area of another.
Since the two figures are similar, it means that there areas will be proportional as their dimensions will be proportional.
Let the dimensions of the smaller figure be k and the larger figure is p times the smaller figure. Then the area of the smaller figure is [tex]k^2[/tex] and the area of the larger figure is [tex](pk)^2[/tex].
Now, given that the area of the larger figure is nine times the area of the smaller figure, this means that:
[tex] \frac{(pk)^2}{k^2} = \frac{9}{1} \\ \\ \frac{p^2k^2}{k^2} =9 \\ \\ p^2=9 \\ \\ p= \sqrt{9} \\ \\ p=3[/tex]
Therefore, the
larger figure must have dimensions that are 3 times the dimensions of the
smaller figure.
