Problem 9 · Areas between parallels
Three regions, one unknown: the similarity ratio $p$.
Integer answer, at most 4 digitsThe picture shows a triangle subdivided into smaller triangles. The area of one of the triangles is given, and we also know that the shaded area equals $180$ cm$^2$. Sides marked in the same way are parallel. What is the area of the triangle marked with a question mark?
Reasoned solution
Key idea: call the big triangle's vertices $L$, $T$, $R$ (left, top, right), $D$ the point on the base, and $S$ the total area. Let $p = \tfrac{LD}{LR}$.
The single-mark segment leaves $D$ parallel to side $TR$, and the double-mark one leaves $D$ parallel to side $LT$. This creates two triangles similar to the big one:
The two grey regions are triangles with bases on side $LT$ (their bases add up to the upper stretch, of length $(1-p) \cdot LT$) and apexes at $D$ and at the point on the right side, both at distance $p \cdot h$ from line $LT$. Together they are worth
Dividing: $\tfrac{1-p}{p} = \tfrac{180}{120} = \tfrac{3}{2}$, so $p = \tfrac{2}{5}$ and $S = \tfrac{120}{p^2} = 750$.