Square ratio
A square is divided into four rectangles and a square. Can you work out the ratio of the side lengths of the rectangles?
Age
14 to 16
Challenge level
Problem
A square is divided into four congruent rectangles and a smaller square, as shown.
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The area of the small square is ${\frac{1}{4}}$ of the area of the whole square.
What is the ratio of the length of a short side of one of the rectangles to the length of a long side?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 1:3
Using lengths
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If the areas are in the ratio $1:4,$ then the sides are in the ratio $1:2$
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$x=s+l,$ and
$\frac12 x+s+s = x\\
\Rightarrow x + 4s = 2x\\
\Rightarrow x = 4s$
So $4s = s + l \Rightarrow 3s = l$ so the ratio is $1:3$.
Using areas
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$(l-s)^2 : (l+s)^2 = 1:4$
$\Rightarrow(l-s)^2\times4 = (l+s)^2$
$\Rightarrow4 = \dfrac{(l+s)^2}{(l-s)^2}$
$\Rightarrow2 = \dfrac{l+s}{l-s}$
$\Rightarrow 2l-2s = l+s$
$\Rightarrow l = 3s$
So the ratio is $1:3$.