The area of a right triangle is half the product of the lengths of its legs, which here are \(\displaystyle A\) feet and \(\displaystyle B\) feet.

Multiply each length by 12 to convert to inches - the lengths become \(\displaystyle 12A\) and \(\displaystyle 12 B\). The area in square inches is therefore

\(\displaystyle \frac{1}{2} (12A )(12 B) = \frac{1}{2} \cdot 12 \cdot 12 \cdot A \cdot B = 72AB\) square inches.

Square \(\displaystyle ABCD\) has area 1,600, so the length of each side is \(\displaystyle \sqrt{1,600 }= 40\).

Since \(\displaystyle AZ = 16\),

\(\displaystyle BZ = AB - AZ = 40 - 16 = 24\)

Therefore, \(\displaystyle BZ > AZ\).

\(\displaystyle \bigtriangleup AZX\) has as its area \(\displaystyle \frac{1}{2} \cdot AX \cdot AZ\); \(\displaystyle \bigtriangleup BZY\) has as its area \(\displaystyle \frac{1}{2} \cdot BX \cdot BZ\).

Since \(\displaystyle BZ > AZ\) and \(\displaystyle AX = BX\), it follows that

\(\displaystyle BX \cdot BZ > AX \cdot AZ\)

and

\(\displaystyle \frac{1}{2} \cdot BX \cdot BZ >\frac{1}{2} \cdot AX \cdot AZ\)

\(\displaystyle \bigtriangleup BZY\) has greater area than \(\displaystyle \bigtriangleup AZX\).

Square \(\displaystyle SUAE\) has area 400, so its common sidelength is the square root of 400, or 20. Therefore, 

\(\displaystyle AR = AE- ER = 20 - 9 = 11\).

The area of a right triangle is half the product of the lengths of its legs.

\(\displaystyle \bigtriangleup SQE\) has legs \(\displaystyle \overline{SQ}\) and \(\displaystyle \overline{SE}\), so its area is

\(\displaystyle \frac{1}{2} \cdot SQ \cdot SE = \frac{1}{2} \cdot 9 \cdot 20 = 90\).

\(\displaystyle \bigtriangleup UAR\) has legs \(\displaystyle \overline{UA}\) and \(\displaystyle \overline{AR}\), so its area is

\(\displaystyle \frac{1}{2} \cdot AR \cdot UA = \frac{1}{2} \cdot 11 \cdot 20 = 110\).

\(\displaystyle \bigtriangleup UAR\) has the greater area.

Opposite sides of a parallelogram have the same measure, so

\(\displaystyle AB = CD\)

\(\displaystyle AM + MB = CD\)

\(\displaystyle 60 + MB = 120\)

\(\displaystyle 60 + MB - 60 = 120 - 60\)

\(\displaystyle MB = 60\)

 Base \(\displaystyle \overline{AM}\) of \(\displaystyle \bigtriangleup AMD\) and base \(\displaystyle \overline{MB}\) of \(\displaystyle \bigtriangleup BMC\) have the same length;  also, as can be seen below, both have the same height, which is the height of the parallelogram.

Therefore, the areas of \(\displaystyle \bigtriangleup AMD\) and \(\displaystyle \bigtriangleup BMC\) have the same area - \(\displaystyle \frac{1}{2} \cdot 60 \cdot h = 30h\).

The perimeter of the triangle - the sum of the lengths of its sides - is

\(\displaystyle 5 + 12 + 13 = 30\) inches. 

3 feet are equivalent to \(\displaystyle 3 \times 12 = 36\) inches, so this is the greater quantity.

One meter is equivalent to 100 centimeters, so 370 meters can be converted to centimeters by multiplying by 100:

\(\displaystyle 370 \times 100 = 37,000\).

370 meters are equal to 37.000 centimeters and is the greater quantity.

One kilometer is equal to 1,000 meters, so convert 4.8 kilometers to meters by multiplying by 1,000:

\(\displaystyle 4.8 \times 1,000 = 4,800\) meters

The two quantities are equal.

A regular pentagon has five sides of equal length. The perimeter, which is the sum of the lengths of these sides, is one yard, which is equal to 36 inches. Therefore, the length of one side is

\(\displaystyle 36 \textrm{ in} \div 5 = 7\frac{1}{5} \textrm{ in}\).

The perimeter can be converted from feet to inches by multiplying by conversion factor 12 {inches per foot): 

\(\displaystyle 8 \textrm{ ft} \times 12 \textrm{ in/ft} = 96 \textrm{ in}\)

A regular hexagon has six sides of equal length. Divide this perimeter by 6 to obtain the length of each side:

\(\displaystyle 96 \textrm{ in} \div 6 = 16 \textrm{ in}\)

(a) A cube has six faces, each a square. Since the surface area of this cube is \(\displaystyle 600 \textrm{ cm}^{2}\), each face has one-sixth this area, or \(\displaystyle 100 \textrm{ cm}^{2}\); the sidelength is the square root of this, or \(\displaystyle 10 \textrm{ cm}\).

(b) The volume of a cube is the cube of its sidelength, so we take the cube root of the volume of this cube to get the sidelength: 

\(\displaystyle \sqrt[3]{1,000} = 10 \textrm{ cm}\)

The cubes have the same sidelength.