Answer to Question #106887 in Geometry for Amy

Question

Rectangle ABCD is cut into four smaller rectangles, as shown in the figure.
The four smaller rectangles have the following properties: (a) the
perimeters of three of them are 11, 16 and 19; (b) the perimeter of the
fourth is neither the biggest nor the smallest of the four. What is the
perimeter of the original rectangle ABCD?

Answer 1

We know that the perimeter of the big rectangle 1/2 the sum of the perimeters of the four smaller rectangles. Let x₁ and x₂ be the sizes of the horizontal cuts and y₁ and y₂ be the sizes of the vertical cuts.

Then 2(x₁ + y₁) = 16,

2(x₁ + y₂) = 11,

2(x₂ + y₁) = 19

and so the perimeter of the mystery rectangle:

 2(x₂ + y₂) = 2(x₂ + y₁) + 2(x₁ + y₂) − 2(x₁ + y₁) = 11 + 19 − 16 = 14

making the total perimeter 30