Diagonal area
A square has area 72 cm$^2$. Find the length of its diagonal.
Problem
A square has area 72 cm$^2$. Find the length of its diagonal.
This problem is taken from the World Mathematics Championships
Student Solutions
Using area
The diagram below shows the square with area 72 cm$^2$ with the diagonal in red. Then 4 copies of the square are stuck together. Notice that the diagonals form another square, which is also shown on its own.
Image
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The square formed by the diagonals occupies half of the area of the 4 blue squares, so it has the same area as 2 blue squares, which is 72 + 72 = 144 cm$^2$.
So a square with side length equal to the red diagonal has area 144 cm$^2$, which means the diagonal must be 12 cm, since 12 $\times$ 12 = 144.
Using Pythagoras' Theorem
The area of the square is $72$ cm$^2$, so theside length is $\sqrt{72}$. This is shown on the diagram below, which also shows the diagonal, marked $d$.
Image
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Applying Pythagoras' Theorem gives