Problem
GEO-B1-M06-P009 Two Midpoints
#9
★★☆☆☆ Level 2 of 5
In triangle \(ABC\), the area is \(64\). Points \(D\) and \(E\) are the midpoints of sides \(AB\) and \(BC\). Find \(S_{BDE}\).
In triangles \(BDE\) and \(BAC\), the two sides from \(B\) are halved.
We have \(BD=\frac{1}{2}BA\) and \(BE=\frac{1}{2}BC\), and the angle at \(B\) is common. The area is reduced by \(2\cdot2=4\) times. Therefore \(S_{BDE}=\frac{64}{4}=16\).
Can be solved by similarity or by \(\frac{1}{2}ab\sin\theta\); for first level, explaining the area scale is enough.