🦘 Math Kangaroo Grade All Felix 1-2 Ecolier 3-4 Benjamin 5-6 Kadett 7-8 Junior 9-10 Student 11-12 ⇄ switch contest
2017 Math Kangaroo

Problem 30

Problem 30 · 2017 Math Kangaroo Stretch
Geometry & Measurement area-fractionarea-decomposition

The parallelogram ABCD has area 1. The two diagonals intersect each other at point M. Another point P lies on the side DC. E is the point of intersection of the segments AP and BD, and F is the point of intersection of the segments BP and AC. What is the area of the quadrilateral EMFP, if the sum of the areas of the triangles AED and BFC is 13?

Figure for Math Kangaroo 2017 Problem 30
Show answer
Answer: D — 112
Show hints
Hint 1 of 2
Use that the parallelogram has area 1 and that its diagonals and the segments cut it into known fractions.
Still stuck? Show hint 2 →
Hint 2 of 2
Express EMFP as the parallelogram minus the surrounding triangles, using the given AED + BFC = 1/3.
Show solution
Approach: area chase with the given triangle sum
  1. Take all areas relative to the whole parallelogram (=1); the segments AP, BP and the diagonals cut it into triangles of fixed fractions.
  2. Removing the triangles around EMFP and using area(AED) + area(BFC) = 1/3 pins down the quadrilateral.
  3. The area of EMFP is 1/12.
Mark: · log in to save
↑