The floor plan for the indoor living space of Mike's cabin is shaded in the diagram. Mike wants to build square observation decks in opposite corners so the living space and the observation decks create a rectangle. The area of
the living space (shaded region) is 196 square feet.

Step-by-step explanation:

Area of the floor plan = Length × Width

                                    = (x + 12)(x + 8)

Area of the living space = Area of the floor plan - Area of the square observation decks

= (x + 12)(x + 8) - 2x²

= x² + 20x + 96 - 2x²

= -x² + 20x + 96

Since, area of the living space is 196 square feet,

-x² + 20x + 96 = 196

-x² + 20x - 100 = 0

x² - 20x + 100 = 0

(x - 10)² = 0

x = 10

Length of the rectangular floor = (x + 12) = 22 ft

Width of the rectangular floor = (x + 8) = 18 ft

Area of each observation deck = x² = 100 square ft

Perimeter of living space = x + x + 8 + 12 + x + x + 8 + 12

                                          = 4x + 40

                                          = 80 ft                    

Perimeter of the rectangle formed by observation deck and living space will be = 2(x + 8) + 2(x + 12)

     = 4x + 40

     = 80 ft

Statement (1) → Yes

Statement (2) → No

Statement (3) → No

Statement (4) → Yes