Problem
GEO-B2-M07-P023 Two Parallel Sections
#23
★★★★★ Level 5 of 5
In triangle \(ABC\), through points \(P,Q\in AC\), lines parallel to \(BC\) are drawn, meeting \(AB\) at \(P_1,Q_1\). It is known that \(AP:PC=1:2\), \(AQ:QC=2:1\). Find \([APP_1]:[AQQ_1]\).
Each small triangle is similar to \(ABC\), and areas are in the ratio of squares of similarity ratios.
We have \(\frac{AP}{AC}=\frac{1}{3}\), \(\frac{AQ}{AC}=\frac{2}{3}\). Therefore \([APP_1]:[ABC]=\frac{1}{9}\), while \([AQQ_1]:[ABC]=\frac{4}{9}\). Hence \([APP_1]:[AQQ_1]=1:4\).
Stronger than basic similarity: the student must move to squared area ratios.