Problem
GEO-B1-M08-P025 A Parallel Through the Diagonal Intersection
#25
★★★★☆ Level 4 of 5
In trapezoid \(ABCD\), bases \(AD\parallel BC\), \(AD=12\), \(BC=6\). The diagonals meet at \(O\). Through \(O\), a line parallel to the bases meets \(AB\) and \(CD\) at \(X\) and \(Y\). Find \(XY\).
First find how point \(O\) divides the diagonals.
Let \(AD=a\), \(BC=b\). From \(AOD\sim COB\), we get \(DO:OB=a:b\). In triangle \(DBA\), segment \(OX\parallel AD\), so \(OX=\frac{ab}{a+b}\). Similarly, in triangle \(DBC\), \(OY=\frac{ab}{a+b}\). Hence \(XY=\frac{2ab}{a+b}\). For \(a=12\), \(b=6\), we get \(XY=8\).
A strong trapezoid problem: similarity is used twice.