Problem 10 · The path inside the equilateral triangle
Horizontals adding up via Thales and five equal slanted legs.
Integer answer, at most 4 digitsTriangle $ABC$ in the figure is equilateral with side 108 cm. The highlighted solid path is obtained by dividing side $AC$ into 5 equal segments; the horizontal sections of the path are parallel to side $AB$. How many centimetres long is the path?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: by Thales, the horizontal at height $k/5$ (from vertex $C$) measures $\tfrac{k}{5} \cdot 108$; the slanted legs follow the side's direction and measure $\tfrac{108}{5}$ each.
The four horizontal sections:
$$\left(\tfrac{1}{5} + \tfrac{2}{5} + \tfrac{3}{5} + \tfrac{4}{5}\right)\cdot 108 = 2 \cdot 108 = 216\ \text{cm}.$$
The five slanted legs (one per fifth of the side): $5 \cdot \dfrac{108}{5} = 108$ cm.
$$216 + 108 = 324\ \text{cm}.$$
Answer: 324