In the rectangle ABCD pictured, M 1 is the midpoint of DC , M 2 the midpoint of AM 1 , M 3 the midpoint of BM 2 and M 4 the midpoint of CM 3 . Determine the ratio of the area of the quadrilateral M 1 M 2 M 3 M 4 to the area of the rectangle ABCD .

Math Kangaroo Student 2015 Problem 27: Geometry Solution

Problem 27 · 2015 Math Kangaroo Stretch

Geometry & Measurement areaarea-fraction

In the rectangle ABCD pictured, M₁ is the midpoint of DC, M₂ the midpoint of AM₁, M₃ the midpoint of BM₂ and M₄ the midpoint of CM₃. Determine the ratio of the area of the quadrilateral M₁M₂M₃M₄ to the area of the rectangle ABCD.

Figure for Math Kangaroo 2015 Problem 27

Show answer

Answer: C — 732

Show hints

Hint 1 of 2

Put the rectangle on coordinates with side lengths 1 and build the midpoints step by step.

Still stuck? Show hint 2 →

Hint 2 of 2

Use the shoelace formula on the four midpoint coordinates.

Show solution

Approach: coordinates plus the shoelace area formula

Take A(0,0), B(1,0), C(1,1), D(0,1); then M₁(½,1), M₂(¼,½), M₃(⅝,¼), M₄(13/16,⅝).
The shoelace formula on M₁M₂M₃M₄ gives area 7/32.
Since the rectangle has area 1, the ratio is 7/32 (C).

Mark: · log in to save