Notes
zig-zag triangles solution

Solution to the Zig-zag Triangles Puzzle

Solution by Similar Triangles and Area of a Triangle

With the points labelled as above, there are multiple similar triangles. The following triangles are all 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: