# Solution to the [[Zig-zag Triangles]] Puzzle +-- {.image} [[ZigzagTriangles.png:pic]] > The yellow triangles are all right-angled. What fraction is orange? =-- ## Solution by [[Similar Triangles]] and [[Area of a Triangle]] +-- {.image} [[ZigzagTrianglesLabelled.png:pic]] =-- With the points labelled as above, there are multiple [[similar triangles]]. The following triangles are all [[right-angled triangles|right-angled]] and share a vertex at $A$, so are all similar: $A K B$, $A J C$, $A I D$, $A H E$, and $A G F$. Since the line segments $A K$, $K J$, $J I$, $I H$, and $H G$ are all the same length, the [[length scale factors]] from $A K B$ to the others are, respectively $2$, $3$, $4$, and $5$. Therefore, $G F$ is five times longer than $K B$, so since triangles $H G F$ and $A K B$ are both right-angled and have the same length bases, the area of $H G F$ is five times that of $A K B$. Similarly, $I H E$ is four times that of $A K B$, $J I D$ is three times, and $K J C$ twice. The total area of the yellow triangles is therefore $5 + 4 + 3 + 2 + 1 = 15$ times that of $A K B$. The total area of $A G F$ is $5^2$ that of $A K B$. So the fraction of the triangle that is orange is: $$ \frac{25 - 15}{25} = \frac{2}{5} $$