Problem
GEO-B2-M10-P011 Homothety in a Triangle
#11
★★★☆☆ Level 3 of 5
In triangle \(ABC\), points \(D\in AB\), \(E\in AC\), with \(DE\parallel BC\) and \(AD:DB=2:3\). Find \([ADE]:[ABC]\).
Triangles \(ADE\) and \(ABC\) are similar with homothety centre \(A\).
Since \(AD:DB=2:3\), we have \(AD:AB=2:5\). From \(DE\parallel BC\), \(\triangle ADE\sim\triangle ABC\) with scale factor \(\frac{2}{5}\). Areas of similar triangles are in the square of the scale factor, so \([ADE]:[ABC]=4:25\).
The problem checks whether the student sees homothety, not only parallel lines.