https://file.aiquickdraw.com/mathbot/file/172641165157633kra3a6.jpg

Let's analyze the image and solve for the perimeter of the figure ABCDE.

### Step 1: Identifying known values and structure
- \( AB = 12 \)
- \( BC = 12 \)
- \( CD = 12 \)
- Angles \( \angle ADE = 30^\circ \) and \( \angle DAE = 30^\circ \) form two 30° triangles.
- \( \angle ABE \) and \( \angle EDC \) are right angles.

### Step 2: Using trigonometry to find the unknown sides
We can focus on triangles \( ADE \) and \( CDE \) to find lengths \( AE \) and \( DE \).

In triangle \( ADE \):
- \( AD = 12 \)
- \( \angle ADE = 30^\circ \)

Using the cosine function:
\[
AE = AD \times \cos(30^\circ) = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3} \approx 10.39
\]

Using the sine function:
\[
DE = AD \times \sin(30^\circ) = 12 \times \frac{1}{2} = 6
\]

### Step 3: Finding the perimeter
The perimeter of ABCDE is the sum of all sides:
\[
P = AB + BC + CD + DE + EA
\]
Substituting the known values:
\[
P = 12 + 12 + 12 + 6 + 10.39 \approx 52.39
\]

Thus, the perimeter of the figure ABCDE is approximately **52.39 units**.

Would you like further clarification or more details on any step? Here are some related questions:

1. How is the cosine and sine of an angle used in solving triangle problems?
2. What is the Pythagorean theorem, and how could it be applied here?
3. Can you explain why the angles in a right triangle sum to 90°?
4. How would changing the angle \( \angle ADE \) affect the perimeter of the figure?
5. How do you calculate the area of triangle \( ADE \)?

**Tip:** For any right triangle, the sine and cosine of 30° are standard values you should memorize: \( \sin(30^\circ) = 1/2 \) and \( \cos(30^\circ) = \sqrt{3}/2 \).

This guide walks through calculating the perimeter of a right triangle figure with 30° angles using trigonometric identities. The problem applies cosine and sine of 30° to find unknown sides, then sums all side lengths for the final perimeter.

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