One yard is equal to three feet, so the length of one side of a square with this perimeter is \(\displaystyle \frac{3}{4}\) feet. The area of the square is \(\displaystyle \frac{3}{4}\times \frac{3}{4} = \frac{9}{16}\) square feet. \(\displaystyle \frac{9}{16} > \frac{1}{2}\), making (a) greater.

Let \(\displaystyle s\) be the sidelength of Square 1. Then the length of a diagonal of this square - which is \(\displaystyle \sqrt{2}\) times this sidelength, or \(\displaystyle s \sqrt{2}\), by the \(\displaystyle 45^{\circ }-45^{\circ }-90^{\circ }\) Theorem - is the same as the diameter of this circle, which, in turn, is equal to the sidelength of Square 2. 

Therefore, Square 1 has area \(\displaystyle A = s^{2}\), and Square 2 has area \(\displaystyle A = \left ( s \sqrt{2 } \right )^{2} = 2s^{2}\), twice that of Square 1.

The area of a square is the square of its sidelength, which here is 12 inches:

\(\displaystyle A = 12^{2} = 144\) square inches.

The area of a rectangle is its length multiplied by its height, which, respectively, are 9 inches and 14 inches:

\(\displaystyle A = 9 \cdot 14 = 126\) square inches.

The square has the greater area.

20 yards converts to \(\displaystyle 20 \times3 = 60\) feet. The area of a square is the square of its sidelength, so the area in square feet is \(\displaystyle 60^{2} = 60 \times 60 = 3,600\) square feet.

The perimeter of a rectangle can be given by the formula

\(\displaystyle P = 2(L+W)\)

Rectangle A has perimeter 2 meters, which is equal to 200 centimeters, and width 25 centimeters, so the length is:

\(\displaystyle 2(L+W) = P\)

\(\displaystyle 2(L+25) = 200\)

\(\displaystyle 2L + 50= 200\)

\(\displaystyle 2L = 150\)

\(\displaystyle L = 75\)

The dimensions of Rectangle A are 75 centimeters and 25 centimeters, so its area is 

\(\displaystyle 75 \times 25 = 1,875\) square centimeters.

The sidelength of a square is one-fourth its perimeter, which here is 

\(\displaystyle \frac{200}{4} = 50\) centimeters; its area is therefore 

\(\displaystyle 50 ^{2} = 2,500\) square centimeters.

The area of Rectangle A is therefore 

\(\displaystyle \frac{1,875}{2,500} \times 100 = 75 \%\)

that of Square B.

Since the ratio is the same regardless of the sidelengths, then for simplicity's sake, assume the sidelength of Square B is 7. The area of Square B is therefore the square of this, or 49.

Then the sidelength of Square A is three-sevenths of 7, or 3. Its area is the square of 3, or 9. 

The ratio of the area of Square B to that of Square A is therefore 49 to 9.

The areas of the squares are:

\(\displaystyle 1^{2} = 1\) square foot

\(\displaystyle 2^{2} = 4\) square feet

\(\displaystyle 3^{2} = 9\) square feet

\(\displaystyle 4^{2}= 16\) square feet

\(\displaystyle 5^{2}= 25\) square feet

Therefore, we are comparing the mean and the median of the set \(\displaystyle \left \{ 1, 4, 9, 16, 25\right \}\).

The mean of this set is the sum divided by 5:

 \(\displaystyle (1 + 4 + 9 + 16 + 25) \div 5 = 55 \div 5 = 11\) 

The median is the middle element after arrangement in ascending order, which is 9.

This makes (A), the mean, greater.

The areas of the squares are:

\(\displaystyle 100 \cdot 100 = 10,000\) square centimeters (one meter being 100 centimeters)

\(\displaystyle 120 \cdot 120 = 14,400\) square centimeters

\(\displaystyle 140 \cdot 140 = 19,600\) square centimeters

\(\displaystyle 140 \cdot 140 = 19,600\) square centimeters

The mean of these four areas is their sum divided by four:

\(\displaystyle (10,000+14,400+19,600+19,600) \div 4\)

\(\displaystyle = 63,600 \div 4 =15,900\) square centimeters.

The median is the mean of the two middle values, or

\(\displaystyle ( 14,400+19,600 ) \div 2 = 34,000 \div 2 = 17,000\) square centimeters.

The median, (B), is greater.

The length of one side of a square is one fourth its perimeter. Since the perimeter of the square is \(\displaystyle 12x+16\), the length of one side is

\(\displaystyle (12x+16) \div 4 = 3x+ 4\)

The area of the square is the square of this sidelength, or

\(\displaystyle ( 3x+ 4)^{2}= (3x)^{2}+ 2 \cdot 3x \cdot 4 + 4 ^{2} = 9x^{2}+24x + 16\)

The area of a square is the square of its sidelength. Therefore, square \(\displaystyle 2^{x}\):

\(\displaystyle \left (2^{x} \right ) ^{2} = 2 ^{2 x}\)