Shaded square
Weekly Problem 41 - 2016
The diagram shows a square, with lines drawn from its centre. What is the shaded area?
The diagram shows a square, with lines drawn from its centre. What is the shaded area?
Age
14 to 16
Challenge level
Problem
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The diagram on the right shows a square with side length $2\text{m}$ and lines drawn to its sides from its centre $O$. The points $A$, $B$, $C$ and $D$ are all on different sides of the square.
The lines $OA$ and $OB$ are perpendicular, as are $OC$ and $OD$.
What is the shaded area?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
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Then, $\angle DOC = 90^\circ = \angle ZOX$, so: $\angle DOZ= 90^\circ - \angle ZOC = \angle COX$. This makes $DOZ$ and $COX$ congruent, as they have the same angles and one pair of corresponding sides the same length.
But then, $CODY$ and $ZOXY$ have the same areas, as they both attach one of these triangles to $COZY$. This is clearly a quarter of the large square.
There are two of these unshaded areas, and both have area one quarter that of the square. This means half the square is shaded, so $\frac{1}{2} \times 2^2 = 2\text{m}^2$.
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Alternatively, extending the lines from $D$ and $C$ past $O$ divides the square into four congruent pieces, since rotation by $90^\circ$ takes each piece to the next. Therefore, each unshaded area is one quarter of the total area of the square.
There are two of these unshaded areas, and both have area one quarter that of the square. This means half the square is shaded, so $\frac{1}{2} \times 2^2 = 2\text{m}^2$.