Problem
GEO-B1-M04-P014 Perimeter of the Midpoint Parallelogram
#14
★★★☆☆ Level 3 of 5
In quadrilateral \(ABCD\), points \(M,N,P,Q\) are the midpoints of sides \(AB,BC,CD,DA\). It is known that diagonals \(AC=11\) and \(BD=15\). Find the perimeter of quadrilateral \(MNPQ\).
Each side of \(MNPQ\) equals half of one diagonal of the original quadrilateral.
By the midline theorem, \(MN=PQ=\frac{1}{2}AC=\frac{11}{2}\), and \(NP=QM=\frac{1}{2}BD=\frac{15}{2}\). Thus the perimeter is \(2\cdot\frac{11}{2}+2\cdot\frac{15}{2}=11+15=26\).
A good task for turning geometry into a short formula after proof.