This Is True For All Squares!

Geometry Level 1

I draw a square.
I label the midpoints of all the sides of this square.
I then connect these four midpoints to form another square.
The area of the original square is $\text{\_\_\_\_\_\_}$ the area of the new square.

Five times Four times Two times Three times

1 solution

Sam Bealing
May 8, 2016

Let the big square have side length $2x$ . By Pythagoras' theorem on one of the triangles in the corner we have the side length of the smaller square is:

$\sqrt{x^2+x^2}=\sqrt{2x^2}=\sqrt{2} x$

The ratio of areas is therefore:

$\dfrac{(2x)^2}{(\sqrt{2} x)^2}=\dfrac{4x^2}{2x^2}=2$

The answer is therefore $\boxed{Two \: times}$

Another way of thinking about this is by splitting the square into 8 triangles (the size of the four triangles in the corner) and the big square covers $8$ triangles and the small square covers $4$ triangles:

$\dfrac{8}{4}=2$

Moderator note:

Both are great approaches :)

This Is True For All Squares!

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