The area of a parallelogram is given by:

\(\displaystyle Area=ba\)

Where \(\displaystyle b\) is the base length and \(\displaystyle a\) is the corresponding altitude. So we can write:

\(\displaystyle Area=t(2t-1)=2t^2-t\)

Base length is \(\displaystyle t\) so the corresponding altitude is  \(\displaystyle \frac{t}{3}\).

The area of a parallelogram is given by:

\(\displaystyle Area=ba\)

Where:

\(\displaystyle b\) is the length of any base
\(\displaystyle a\) is the corresponding altitude

So we can write:

\(\displaystyle 12=t\times \frac{t}{3}\Rightarrow t\times t=12\times 3\Rightarrow t^2=36\Rightarrow t=6\)

\(\displaystyle t\times t=12\times 3\)

\(\displaystyle t^{2}=36\)

\(\displaystyle t=6\)

Let \(\displaystyle D\) be the length of the longer diagonal. Then the shorter diagonal has length 40% of this. Since 40% is equal to \(\displaystyle \frac{40}{100 } = \frac{40 \div 20 }{100 \div 20 } = \frac{2}{5}\), 40% of \(\displaystyle D\) is equal to \(\displaystyle \frac{2}{5}D\).

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up, and solve for \(\displaystyle D\), in the equation:

\(\displaystyle \frac{1}{2} \cdot \frac{2}{5}D \cdot D = K\)

\(\displaystyle \frac{1}{5}D^{2}= K\)

\(\displaystyle 5\cdot \frac{1}{5}D^{2}=5\cdot K\)

\(\displaystyle D^{2}=5K\)

\(\displaystyle D =\sqrt{5K}\)

Let \(\displaystyle D\) be the length of the longer diagonal in yards. Then the shorter diagonal has length two-thirds of this, or \(\displaystyle \frac{2}{3}D\).

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up the following equation and solve for \(\displaystyle D\):

\(\displaystyle \frac{1}{2} \cdot \frac{2}{3}D \cdot D = Q\)

\(\displaystyle \frac{1}{3}D ^{2} = Q\)

\(\displaystyle 3\cdot \frac{1}{3}D ^{2} = 3 \cdot Q\)

\(\displaystyle D ^{2} = 3Q\)

\(\displaystyle D = \sqrt{3Q}\)

To convert yards to inches, multiply by 36:

\(\displaystyle \sqrt{ 3Q } \times 36 = 36\sqrt{ 3Q }\)

Let \(\displaystyle D\) be the length of the shorter diagonal. If the longer diagonal is 20% longer, then it measures 120% of the length of the shorter diagonal; this is 

\(\displaystyle \frac{120}{100} = \frac{120 \div 20}{100 \div 20} = \frac{6}{5}\)

of \(\displaystyle D\), or \(\displaystyle \frac{6}{5}D\).

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up an equation and solve for \(\displaystyle D\):

\(\displaystyle \frac{1}{2} \cdot \frac{6}{5}D \cdot D = N\)

\(\displaystyle \frac{3}{5}D^{2} = N\)

\(\displaystyle \frac{5}{3} \cdot \frac{3}{5}D^{2} = \frac{5}{3} \cdot N\)

\(\displaystyle D^{2} = \frac{5N}{3}\)

\(\displaystyle D =\sqrt{ \frac{5N}{3}}\)

A quadrilateral is any two-dimensional shape with  \(\displaystyle 4\) sides. The only shape listed that does not have \(\displaystyle 4\) sides is a triangle. 

The only difference between a rectangle and a square is their side lengths. A square has to have \(\displaystyle 4\) equal side lengths, but the opposite side lengths of a rectangle only have to be equal. 

Out of the choices given, the only characteristic used to describe shapes is the number of sides. A triangle has \(\displaystyle 3\) sides and a rectangle has \(\displaystyle 4\) sides. 

By definition, the only two quadrilaterals that have to have \(\displaystyle 4\) right angles, are the square and the rectangle. 

A quadrilateral is a \(\displaystyle 4\) sided shape. The only shape listed that does not have \(\displaystyle 4\) sides is a hexagon, which has \(\displaystyle 6\) sides.