Similar rectangles
Can you find the missing length?
Age
14 to 16
Challenge level
The smaller of two similar rectangles has height $2$ units; the larger rectangle has length $6$ units.
If one rectangle has twice the area of the other, find the length of the smaller rectangle.
This problem is taken from Tony Gardiner's 'Extension Mathematics Gamma' book.
Call the length of the smaller rectangle $x$.
If the area of the smaller rectangle is $2x$, the area of the larger rectangle is $4x$, and the height of the larger rectangle is $4x/6$. The height:length ratio of both rectangles must be the same, since they are similar.
So $\frac{2}{x} = \frac{4x}{36}$
$x^{2} = 18$
$x=\sqrt{18}$
Alternatively, call the height of the larger rectangle $y$.
Then, comparing the areas, $4x = 6y$
By similarity, $\frac{y}{2} = \frac{6}{x}$
i.e. $y = \frac{12}{x}$
Then by substitution, $4x = \frac{72}{x}$
$x^{2} = 18$
$x = \sqrt{18}$