The formula to find the area of a trapezoid is

\(\displaystyle \text{Area}=\frac{1}{2}(base_1+base_2)(height)\).

Substitute in the values for the area, a base, and the height. Then solve for \(\displaystyle c\).

\(\displaystyle 60=\frac{1}{2}(12+c)6\)

\(\displaystyle 120=(12+c)6\)

\(\displaystyle 12+c=20\)

\(\displaystyle c=8\)

The formula to find the area of a trapezoid is

\(\displaystyle \text{Area}=\frac{1}{2}(base_1+base_2)(height)\).

Substitute in the values for the area, a base, and the height. Then solve for \(\displaystyle d\).

\(\displaystyle 210=\frac{1}{2}(18+d)14\)

\(\displaystyle 420=(18+d)14\)

\(\displaystyle 18+d=30\)

\(\displaystyle d=12\)

Start by drawing in the height, \(\displaystyle CE\), to form a right triangle.

Use the Pythagorean Theorem to find the length of \(\displaystyle CE\).

\(\displaystyle 8^2+(CE)^2=10^2\)

\(\displaystyle 64+(CE)^2=100\)

\(\displaystyle (CE)^2=36\)

\(\displaystyle CE=6\)

Now that we have the height, plug in the given information into the formula to find the area of the trapezoid.

Keep in mind that \(\displaystyle AE+ED=AD\).

\(\displaystyle \text{Area}=\frac{1}{2}(base_1+base_2)(height)\)

\(\displaystyle 96=\frac{1}{2}(12+AE+8)(6)\)

\(\displaystyle \frac{1}{2}(AE+20)=16\)

\(\displaystyle AE+20=32\)

\(\displaystyle AE=12\)

The question asks you to find the length of \(\displaystyle AD\).

\(\displaystyle AD=AE+ED\)

\(\displaystyle AD=12+8=20\)

First, draw in the height \(\displaystyle CE\).

First, find \(\displaystyle CE\) by using \(\displaystyle \text{sine}\).

\(\displaystyle \sin 47=\frac{CE}{8}\)

\(\displaystyle CE=8 \sin 47=5.85\)

Now, plug in the values for area, height, and one base to find the length of the second base, \(\displaystyle AD\).

\(\displaystyle \text{Area}=\frac{1}{2}(base_1+base_2)(height)\)

\(\displaystyle 56=\frac{1}{2}(7+AD)(5.85)\)

\(\displaystyle \frac{1}{2}(7+AD)=9.57\)

\(\displaystyle 7+AD=19.15\)

\(\displaystyle AD=12.15\)

First, draw in the height.

First, find the height by using \(\displaystyle \text{sine}\).

\(\displaystyle \sin 62=\frac{height}{8}\)

\(\displaystyle height=8\sin 62=7.06\)

Now, plug in the values for area, height, and one base to find the length of the second base, \(\displaystyle AD\).

\(\displaystyle \text{Area}=\frac{1}{2}(base_1+base_2)(height)\)

\(\displaystyle 244=\frac{1}{2}(12+AD)(7.06)\)

\(\displaystyle \frac{1}{2}(12+AD)=34.56\)

\(\displaystyle 12+AD=69.12\)

\(\displaystyle AD=57.12\)

First, draw in the height.

First, find the height by using \(\displaystyle \text{sine}\).

\(\displaystyle \sin 58=\frac{height}{10}\)

\(\displaystyle height=10\sin 58=8.48\)

Now, plug in the values for area, height, and one base to find the length of the second base, \(\displaystyle AD\).

\(\displaystyle \text{Area}=\frac{1}{2}(base_1+base_2)(height)\)

\(\displaystyle 124=\frac{1}{2}(14+AD)(8.48)\)

\(\displaystyle \frac{1}{2}(14+AD)=14.62\)

\(\displaystyle 14+AD=29.25\)

\(\displaystyle AD=15.25\)

Quadrilateral \(\displaystyle ABCD\) has at least one pair of parallel sides, \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{CD}\). The figure is, by definition, a parallelogram if and only if \(\displaystyle \overline{AD} \parallel \overline{BC}\), and, by definition, a trapezoid if and only if \(\displaystyle \overline{AD} \nparallel \overline{BC}\). Opposite sides of a parallelogram are also congruent; since \(\displaystyle \overline{AD} \ncong \overline{BC}\), Quadrilateral \(\displaystyle ABCD\) is not a parallelogram. It is therefore a trapezoid.

We can apply the area formula here.

\(\displaystyle A = \frac{1}{2} h \left ( B+b\right )\)

\(\displaystyle 640 = \frac{1}{2}\cdot 20 \left ( B+b\right )\)

\(\displaystyle 640 = 10 \left ( B+b\right )\)

\(\displaystyle B + b = 64\)

The sum of the bases must be 64 inches. We check each one of these choices, except for 32 inches and 32 inches, which can be eliminated as the bases cannot be of the same length.

\(\displaystyle 27+37=64\)

\(\displaystyle 25+35=60\)

\(\displaystyle 18+44=62\)

\(\displaystyle 2+60=62\)

Only 27 and 37 have 64 as a sum, so this is the correct choice.

A kite is a geometric shape that has two sets of equivalent adjacent sides. In order for two kites to be similar their sides must have the same ratios. 

Since, the given kite has side lengths \(\displaystyle 4\) and \(\displaystyle 12\), they have the ratio of \(\displaystyle 4:12=1:3\).

Therefore, find the side lengths that have a ratio of \(\displaystyle 1:3\).

The only answer choice with this ratio is: \(\displaystyle 8:24=1:3\)

A kite is a geometric shape that has two sets of equivalent adjacent sides. In order for two kites to be similar their sides must have the same ratios. 

The given side lengths for the kite are \(\displaystyle 6\) and \(\displaystyle 24\), which have the ratio of \(\displaystyle 1:4\). 

The only answer choice with the same relationship between side lengths is: \(\displaystyle 5\) and \(\displaystyle 20\), which has the ratio of \(\displaystyle 1:4.\)