To find the area of a rhombus, we will use the following formula:

$\displaystyle A = \frac{pq}{2}$

where p and q are the lengths of the diagonals of the rhombus.

Now, we know one diagonal has a length of 12in.  We also know the other diagonal is two times the first diagonal.  Therefore, the second diagonal is 24in.

Knowing this, we can substitute into the formula.  We get

$\displaystyle A = \frac{12\text{in} \cdot 24\text{in}}{2}$

$\displaystyle A = \frac{288\text{in}^2}{2}$

$\displaystyle A = 144\text{in}^2$

Like any polygon, the perimeter of the rhombus is the total distance around the outside, which can be found by adding together the length of each side. In the case of a rhombus, all four sides have the same length, i.e.

$\displaystyle Perimeter=4a$,

where $\displaystyle a$ is the length of each side. 

$\displaystyle Perimeter=4a\Rightarrow 36=4a\Rightarrow a=9\ in$

The area of a rhombus can be determined by the following formula:

$\displaystyle Area=s^2sin\alpha$,

where $\displaystyle s$ is the length of any side and $\displaystyle \alpha$ is any interior angle. 

$\displaystyle Area=s^2sin\alpha\Rightarrow 9\sqrt{2}=s^2\times sin45^{\circ}\Rightarrow 9\sqrt{2}=s^2 \times \frac{\sqrt{2}}{2}$

$\displaystyle \Rightarrow s^2=18\Rightarrow s=\sqrt{18}=\sqrt{9\times 2}=3\sqrt{2}\ cm$

A rhombus has four sides of equal measure, so the perimeter of a rhombus with sidelength 2 yards is $\displaystyle 2 \times 4 = 8$ yards.

$\displaystyle 8 \textrm{ yd} = 8 \times 3 = 24 \textrm{ ft} = 24 \times 12 = 288 \textrm{ in}$

Also, 

$\displaystyle 8 \textrm{ yd} = 24 \textrm{ ft} =\frac{ 24}{5,280} = \frac{1}{220} \textrm{ mi}$

The only choice that agrees with any of these measures is 288 inches, the correct choice.

A rhombus has four sides of equal measure, so the perimeter of a rhombus with sidelength $\displaystyle y + 12$ is $\displaystyle 4 (y + 12) = 4y + 48$.

The total sum of the interior angles of a quadrilateral is $\displaystyle 360$ degrees. In this problem, we are only considering half of the interior angles:

$\displaystyle \frac{360}{2}=180$

$\displaystyle 35+x=180$

$\displaystyle x=145$

The diagonals of a rhombus bisect the angles.

The angle bisected must be supplementary to the $\displaystyle 122^{\circ }$ angle since they are consecutive angles of a parallelogram; therefore, that angle has measure $\displaystyle \left (180 - 122 \right ) ^{\circ } = 58^{\circ }$, and $\displaystyle x$ is half that, or $\displaystyle 29^{\circ }$.

The total area of the rectangle is

$\displaystyle \left [ 3- (-6) \right ] \times\left [ 4- (-2) \right ] = 9 \times 6 = 54$.

The area of the portion of the rectangle in Quadrant I is 

$\displaystyle \left ( 3- 0 \right ) \times\left ( 4- 0 \right) = 3 \times 4 = 12$.

Therefore, the portion of the rectangle in Quadrant I is

 $\displaystyle \frac{12}{54} = \frac{2}{9} = \frac{2}{9} \times 100 \% = 22 \frac{2}{9} \%$.

The sidelength of a square with area $\displaystyle 225$ square centimeters is $\displaystyle \sqrt{225} = 15$ centimeters; its perimeter, as well as that of the rectangle, is therefore $\displaystyle 15 \times 4 = 60$ centimeters. 

Using the formula for the perimeter of a rectangle, substitute $\displaystyle L = 21$ and solve for $\displaystyle W$ as follows:

$\displaystyle 2L + 2W = P$

$\displaystyle 2 \cdot 21 + 2W = 60$

$\displaystyle 42 + 2W = 60$

$\displaystyle 42 -42 + 2W = 60-42$

$\displaystyle 2W = 18$

$\displaystyle 2W \div 2 = 18\div 2$

$\displaystyle W = 9$

In a rectangle in which the width is x while the length is 4x, the first step is to solve for x. If $\displaystyle 3x=9$, the value of x can be found by dividing each side of this equation by 3. 

Doing so gives us the information that x is equal to 3. 

Thus, the area is equal to:

$\displaystyle Area=x\cdot4x$

$\displaystyle Area=3\cdot4\cdot3$

$\displaystyle Area=36$