https://file.aiquickdraw.com/mathbot/file/1727346936105gomxsioi.jpg

We have a right trapezoid in the figure with the following known values:

- The base \( AB = 11 \, \text{cm} \)
- The height \( AC = 8 \, \text{cm} \)
- Angle \( \angle ABD = 60^\circ \)

### Step 1: Find the length of side \( BD \)
Since \( \triangle ABD \) is a right triangle (because \( \angle ABD = 60^\circ \)), we can use trigonometry to find \( BD \), which is the hypotenuse of the triangle \( ABD \).

Using the cosine function for \( \angle ABD = 60^\circ \):
\[
\cos(60^\circ) = \frac{AD}{BD}
\]
We know \( AD \), the adjacent side to the angle, is \( 8 \, \text{cm} \). So:
\[
\cos(60^\circ) = \frac{8}{BD} \implies BD = \frac{8}{\cos(60^\circ)}
\]
\[
\cos(60^\circ) = 0.5 \implies BD = \frac{8}{0.5} = 16 \, \text{cm}
\]

### Step 2: Find the length of side \( CD \)
We can use the sine function to find \( CD \), which is opposite to \( \angle ABD \):
\[
\sin(60^\circ) = \frac{CD}{BD} \implies CD = BD \times \sin(60^\circ)
\]
\[
CD = 16 \times \sin(60^\circ) = 16 \times 0.866 = 13.856 \, \text{cm}
\]

### Step 3: Calculate the area of the trapezoid
The formula for the area of a trapezoid is:
\[
\text{Area} = \frac{1}{2} \times (AB + CD) \times h
\]
Where:
- \( AB = 11 \, \text{cm} \)
- \( CD = 13.856 \, \text{cm} \)
- \( h = AC = 8 \, \text{cm} \)

Substituting the values:
\[
\text{Area} = \frac{1}{2} \times (11 + 13.856) \times 8
\]
\[
\text{Area} = \frac{1}{2} \times 24.856 \times 8 = 99.424 \, \text{cm}^2
\]

### Step 4: Calculate the perimeter of the trapezoid
The perimeter is the sum of all sides:
\[
\text{Perimeter} = AB + AC + CD + BD
\]
\[
\text{Perimeter} = 11 + 8 + 13.856 + 16 = 48.856 \, \text{cm}
\]

### Final Results:
- **Area**: \( 99.42 \, \text{cm}^2 \)
- **Perimeter**: \( 48.86 \, \text{cm} \)

Would you like further details or clarification on any step?

Here are some related questions:
1. How do we derive the formula for the area of a trapezoid?
2. How does the sine and cosine functions help in solving right triangles?
3. What are some other methods to find the height of a trapezoid if it's unknown?
4. What is the difference between a right trapezoid and an isosceles trapezoid?
5. How can we calculate the angles of a non-right trapezoid?

**Tip:** In a right triangle, the sine, cosine, and tangent ratios are the most helpful for finding unknown sides or angles when one angle is known.

Find the area and the perimeter of the right trapezoid shown in the figure. Given: base AB = 11 cm, height AC = 8 cm, and angle ∠ABD = 60°.

Calculate the area and perimeter of a right trapezoid with base AB = 11 cm, height AC = 8 cm, and angle ∠ABD = 60°. The solution includes using trigonometric functions and geometry principles. Suitable for Grades 9-10 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation