In this diagram, not drawn to scale, figures I and III are equilateral triangular regions with areas of 32√3 and 8√3 square inches, respectively. Figure II is a square region with area 32 square inches. Let the length of segment AD be decreased by 12.5% of itself, while the lengths of AB and CD remain unchanged. What’s the percent decrease in the area of the square?

- 12.5
- 25
- 50
- 75
- 87.5

The area of an equilateral triangle of side *s* is (*s*^{2}√3)/4.

In I, (*x*^{2}√3)/4 = 32√3 →*x*^{2}/4 = 32 → *x*^{2} = 128 →

*x* = 8√2.

In II, *y*^{2} = 32 → *y* = 4√2.

In III, (*z*^{2}√3)/4 = 8√3 → *z*^{2}/4 = 8 → *z*^{2} = 32 → *z* = 4√2.

So AD = 8√2 + 4√2 + 4√2 = 16√2.

AD decreases by 12.5%, or 1/8, of itself, so it becomes 14√2. AB and CD are unchanged, so BC shrinks from 4√2 to 2√2.

The area of the square is now (2√2)^{2} = 8, down from 32, a loss of 75%.

The answer is (d).