Question

A diagram shows Parallelogram ABCD with angle A, 30 degrees; side AD, 6; side DC, 10; and a perpendicular dashed line from D to point E on side AB.
Question
Find the area of parallelogram A B C D, where D C equals 10 centimeters, A D equals 6 centimeters, and angle A equals 30°.

Answer options with 5 options
A.
60 centimeters squared

B.
42 centimeters squared

C.
32 centimeters squared

D.
,superscript, ,baseline,30 centimeters squared

E.
30 StartRoot 3 EndRoot centimeters squared

Answer 1

To find the area of parallelogram ABCD, we can use the formula for the area of a parallelogram, which is given by:

\[
\text{Area} = \text{base} \times \text{height}
\]

In this case, we can take side DC (which is 10 cm) as the base. To find the height, we need to determine the perpendicular distance from point D to line AB.

Given:
- Side \(AD = 6 \text{ cm}\)
- Side \(DC = 10 \text{ cm}\)
- Angle \(A = 30^\circ\)

We can find the height \(h\) using the sine function from trigonometry as follows:

\[
h = AD \cdot \sin(\angle A)
\]

Plugging in the values:

\[
h = 6 \cdot \sin(30^\circ)
\]

Since \(\sin(30^\circ) = \frac{1}{2}\):

\[
h = 6 \cdot \frac{1}{2} = 3 \text{ cm}
\]

Now, we can calculate the area of the parallelogram:

\[
\text{Area} = \text{base} \times \text{height} = DC \cdot h = 10 \cdot 3 = 30 \text{ cm}^2
\]

So, the area of parallelogram ABCD is **30 cm²**.

Thus, the correct answer is:
**E. 30 centimeters squared.**