# Parallelogram Area +-- {.image} [[ParallelogramArea.png:pic]] > $a + b$ is a right angle. What's the area of the parallelogram? =-- ## Solution by [[Similar Triangles]] and [[Pythagoras' Theorem]] +-- {.image} [[ParallelogramAreaLabelled.png:pic]] =-- In the diagram above point $C$ is on the continuation of $A B$ so that angle $D \hat{C} B$ is a [[right-angle]]. Since [[angles in a triangle]] add up to $180^\circ$, angle $B \hat{D} C$ is therefore $90^\circ - b$ which is $a$. Triangle $A C D$ is also a [[right-angled triangle]] with angle $a$ and so triangles $A C D$ and $D C B$ are [[similar]]. Since $A D$ has length $10$ and $B D$ has length $5$, triangle $D C A$ is double triangle $B C D$. Let $x$ be the length of $B C$. Then $D C$ has length $2 x$ and $A C$ has length $4 x$, so $A B$ has length $3 x$. The area of the parallelogram is then $3 x \times 2 x = 6 x^2$. Applying [[Pythagoras' theorem]] to triangle $D B C$ yields the equation: $$ x^2 + (2 x)^2 = 5^2 $$ and so $x^2 = 5$. Therefore, the area is $30$.