Problem
GEO-B1-M08-P013 Equal Areas in a Trapezoid
#13
★★★★☆ Level 4 of 5
In trapezoid \(ABCD\), bases \(AD\parallel BC\), and the diagonals meet at point \(O\). Prove that \(S_{AOB}=S_{COD}\).
Compare large triangles \(ABD\) and \(ACD\), then subtract the common part.
Triangles \(ABD\) and \(ACD\) have common base \(AD\). The heights from \(B\) and \(C\) to \(AD\) are equal because \(BC\parallel AD\). Hence \(S_{ABD}=S_{ACD}\). Subtracting the common area \(S_{AOD}\), we get \(S_{AOB}=S_{COD}\).
A real mixed proof: trapezoid, diagonals, and areas.