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Problem 1153

# In parallelogram $$\mathrm{ABCD}, \mathrm{AB}=8$$ inches, $$\mathrm{AD}=6$$ inches, and angle $$\mathrm{A}=30^{\circ}$$. Find the number of square inches in the area of the parallelogram.

Expert verified
The area of parallelogram ABCD is $$24$$ square inches.
See the step by step solution

## Step 1: Sketch the parallelogram

Draw the parallelogram ABCD with sides AB and AD given, and the angle A between the two sides. Label the sides AB as 8 inches and AD as 6 inches, and the angle A as 30°.

## Step 2: Find the height of the parallelogram

We need to find the height of the parallelogram, which is the perpendicular distance between the opposite sides (say, AB and CD). To find this height, we can use trigonometry. Let's call the height h. Since we are given that angle A is 30°, we can relate the height, side AD, and angle A, using the sine function: $h = \text{AD} \times \sin(\text{angle A})$

## Step 3: Plug in the given values

Now, substitute the given values for side AD (6 inches) and angle A (30°) into the equation: $h = 6 \times \sin(30°)$

## Step 4: Calculate the height

Calculate the height h: $h = 6 \times 0.5 = 3$ The height of the parallelogram is 3 inches.

## Step 5: Find the area of the parallelogram

Now that we have the height. We can calculate the area of the parallelogram using the formula: $\text{Area} = \text{base} \times \text{height}$ In this case, base is side AB, which is given as 8 inches.

## Step 6: Plug in the values in the formula

Now, substitute the given value for the base (8 inches) and the height we found earlier (3 inches) into the formula: $\text{Area} = 8 \times 3$

## Step 7: Calculate the area

Compute the area: $\text{Area} = 24$ The area of parallelogram ABCD is 24 square inches.

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