# Three Equal Area Rectangles +-- {.image} [[ThreeEqualAreaRectangles.png:pic]] > All three of these rectangles have the same area. What is it? =-- ## Solution by [[Pythagoras' Theorem]] +-- {.image} [[ThreeEqualAreaRectanglesLabelled.png:pic]] =-- In the above diagram, consider the [[right-angled triangle]] $A B C$. Let $x$ be the length of $A D$, then $A B$ has length $x + 10$ and $C B$ has length $x - 10$. Let $y$ be the length of $A C$. Then applying [[Pythagoras' theorem]] to triangle $A B C$ yields: $$ y^2 = (x + 10)^2 + (x - 10)^2 = 2 x^2 + 200 $$ As the rectangles have the same area, also $7 y = 10 x$ so $y = \frac{10}{7} x$. Putting this into the above equation, $$ \frac{100}{49} x^2 = 2 x^2 + 200 $$ which simplifies to $\frac{2}{49} x^2 = 200$, and this simplifies further to $x = 70$ and so $y = 100$. The area of the rectangles is therefore $700$.