Pentagonal Problem

Geometry Level 3

Let $ABCDE$ be a convex pentagon. If $P$ , $Q$ , $R$ and $S$ are the respective centroids of the triangles $ABE$ , $BCE$ , $CDE$ and $DAE$ , then which of the following is true :

$A$ : $PQRS$ is a parallelogram and its area is $2/9$ of that of $ABCD$ .

$B$ : $PQRS$ is a rectangle and its area is $2/9$ of that of $ABCD$ .

$C$ : $PQRS$ is a parallelogram and its area is $1/3$ of that of $ABCD$ .

$D$ : $PQRS$ is a rectangle and its area is $1/3$ of that of $ABCD$ .

B C D A

1 solution

Marta Reece
May 1, 2016

Take the centers of the sides of the convex but otherwise general quadrilateral ABCD. Let's call them EFGH. These four points will form a parallelogram with an area half that of the area of ABCD. Points P, Q, R, and S are each 2/3 of the way between point E and the appropriate point E, F, G, or H. This means that PQRS is also a parallelogram and its area is 2/3 squared times the area of EFGH.

$\frac {1}{2} ( \frac {2}{3} )^2 = \frac {2}{9}$

(This is just a bare bones outline of a solution, but since there wasn't any...)

Pentagonal Problem

1 solution

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