Midpoints of the Sides (Between the Lines of Geometric Brilliance)

Geometry Level 3

Let $ABCD$ be a quadrilateral. Let $M_1 , M_2, M_3,$ and $M_4$ be the midpoints of sides AB, BC, CD, DA, respectively. Given that $M_1M_2 = 5$ and $M_2M_3 = 25$ , find the perimeter of quadrilateral $M_1M_2M_3M_4$ .

The answer is 60.

1 solution

Alan Yan
Aug 10, 2015

Lemma: The midpoints of a quadrilateral form a parallelogram.

Proof: $\vec{M_1} = \frac{\vec{A}+\vec{B}}{2} ; \vec{M_2} = \frac{\vec{B} + \vec{C}}{2} ; \vec{M_3} = \frac{\vec{C}+\vec{D} }{2} ; \vec{M_4} = \frac{\vec{D}+\vec{A}}{2}$ Through simple computations, it is easy to proof that $\vec{M_1M_2} = \vec{M_3M_4}$

Therefore the side lengths of the parallelogram must be 5, 25, 5, 25 which implies a perimeter of 60.

Midpoints of the Sides (Between the Lines of Geometric Brilliance)

The answer is 60.

1 solution

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