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Chapter 5: Problem 14

Find the area of a parallelogram that has pairs of sides of lengths 5 and 11 , with a \(28^{\circ}\) angle between two of those sides.

Short Answer

Expert verified
The area of the parallelogram with side lengths 5 and 11, and a \(28^{\circ}\) angle between the sides is approximately 25.95 square units.

Step by step solution

01

Understand the problem and draw a diagram.

In this problem, we are asked to find the area of a parallelogram with given side lengths of 5 and 11, and an angle of 28掳 between the two sides. Before we proceed, let's draw a diagram to visualize the problem. When drawing the parallelogram, label the vertices A, B, C, and D, such that AB = 5, BC = 11, and the angle 鈭燗BC is 28掳.
02

Calculate the height of the parallelogram using trigonometry.

To find the area of the parallelogram, we need to find the height, which is the perpendicular distance between the base and the top of the parallelogram. In our case, base is segment BC or side length 11. Let's now find the height of the parallelogram by using trigonometry. Draw a perpendicular from point A to line CD, and label the intersection point E. The triangle ABE is a right triangle, and AE is the height we are looking for. Notice that we can use the sine function to relate the side lengths of triangle ABE and the angle, 28掳: \[\sin(28^{\circ}) = \frac{AE}{AB}\] Now we can solve for AE, the height of the parallelogram:
03

Solve for the height (AE).

Plug in the given values and solve for AE in the sine equation: \[\sin(28^{\circ}) = \frac{AE}{5}\] AE = 5 脳 sin(28掳) AE 鈮� 2.359 Now that we have the height of the parallelogram, we can find its area.
04

Find the area of the parallelogram using the area formula.

The formula for the area of a parallelogram is given by: Area = base 脳 height In our problem, the base is BC, and the calculated height is AE. So, we have: Area 鈮� 11 脳 2.359 Area 鈮� 25.95 Therefore, the area of the parallelogram is approximately 25.95 square units.

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