The formula for the area of a rhombus is

\(\displaystyle Area=\frac{d_{1}d_{2}}{2}\),

where \(\displaystyle d_{1}\) is the length of one diagonal and \(\displaystyle d_{2}\) is the length of the other diagonal.

\(\displaystyle Area=\frac{d_{1}d_{2}}{2}=\frac{(t-1)(t+1)}{2}=\frac{t^2-1}{2}\)

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. 

\(\displaystyle Area=s^2sin\alpha=4^2\times sin30^{\circ}=16\times \frac{1}{2}=8\ cm^2\)

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

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