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# Kite in a Square

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Age 14 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Viola from St George's British International School, Rome in Italy sent in this elegant method:

However, Viola has assumed that the vertices of the kite are $\frac12$ and $\frac23$ of the way up the whole square - which might not be true! Zach from Pate's Grammar School in England and Raquel from IES Maximo Laguna in Spain both proved that it *is* true using coordinates. Below is Raquel's proof that the vertices of the kite are $\frac12$ and $\frac23$
of the way up the whole square:

Raquel used this fact to complete a different elegant method:

Zach also used coordinates, but Zach worked with a square with sides 10 units long, instead of 1 unit long. Zach also used a different method. Zach began by defining some lengths as $X, Y$ and $Z$:

Zach went on to find equations for the lines and line segments. Be careful - sometimes $X$ and $Y$ refer to the coordinates, and sometimes they refer to the lengths defined above.

Rishik K used the three appoaches linked in the problem to compare three different methods. This is Rishik's work:

I enjoyed solving this problem using all the approaches and I believe that they were all smart and interesting methods to work out the shaded area. But I liked the similar figure approach because it was the simplest out of the three and the quote “*Mathematics is for lazy people*” clearly explains the reason I liked this certain approach.

Pythagoras Approach

Coordinates approach

Similar figures approach