Problem
GEO-B1-M06-P022 Height Along a Cevian
#22
★★★★☆ Level 4 of 5
Triangle \(ABC\) has area \(120\). Point \(D\) lies on \(BC\). Point \(E\) lies on \(AD\), with \(AE:ED=3:2\). Find \(S_{BCE}\).
You need the fraction \(ED:AD\), not \(AE:AD\).
Triangles \(BCE\) and \(BCA\) have common base \(BC\). The height of point \(E\) to \(BC\) is \(ED:AD\) of the height of point \(A\), because \(D\) lies on \(BC\). From \(AE:ED=3:2\), we get \(ED:AD=2:5\). Therefore \(S_{BCE}=\frac{2}{5}\cdot120=48\).
Checks precision: the height is proportional to distance to the base, namely segment \(ED\).