# Three Aligned Triangles +-- {.image} [[ThreeAlignedTriangles.png:pic]] > Given the areas of the three equilateral triangles, find the shaded area. =-- ## Solution by [[Lengths in Equilateral Triangles]] and [[Area Scale Factor]] +-- {.image} [[ThreeAlignedTrianglesLabelled.png:pic]] =-- Let $x$ be half the side length of the smallest triangle. Using the relationships between [[lengths in an equilateral triangle]], the length of $E B$ is $\sqrt{3} x$. The [[area scale factor|area scale factors]] to the other triangles are $9$ and $4$, so the length [[scale factor|scale factors]] are $3$ and $2$. Therefore $A B$ has length $4 x$ and $B C$ has length $3 x$, while $F A$ has length $3 \sqrt{3} x$ and $D C$ has length $2 \sqrt{3} x$. The area of the shaded triangle can be found by taking the area of [[trapezium]] $F D C A$ and subtracting the areas of the trapezia $F E B A$ and $E D C B$. This is given by: $$ \frac{1}{2} 7 x (3 \sqrt{3} x + 2\sqrt{3} x) - \frac{1}{2} 4 x (3 \sqrt{3} x + \sqrt{3} x) - \frac{1}{2} 3 x (2 \sqrt{3} x + \sqrt{3} x) = 5 \sqrt{3} x^2 $$ The area of the smallest equilateral triangle is given by $\sqrt{3} x^2$, so this is equal to $1$. Hence the area of the shaded region is $5$.