https://file.aiquickdraw.com/mathbot/file/1727973553144wglldoug.jpg

The given image shows an equilateral triangle with an inscribed circle, and the dimensions of the triangle are provided.

To find the perimeter of the polygon (which is an equilateral triangle), we'll use the following observations:

1. **Side of the triangle**: The distance from the center of the circle to one of the sides is given as 5 inches (the radius of the circle). 
2. The height of the triangle is given as 10 inches, and the perpendicular distance from the center of the circle to a vertex of the triangle is 6 inches.

Using the geometric properties of an equilateral triangle, we can now solve the problem.

### Step 1: Side Length Calculation
In an equilateral triangle, the relationship between the height \( h \) and the side length \( s \) is given by:
\[
h = \frac{\sqrt{3}}{2} s
\]
Given that the height of the triangle is 10 inches, we can solve for the side length \( s \):
\[
10 = \frac{\sqrt{3}}{2} s
\]
Multiplying both sides by 2:
\[
20 = \sqrt{3} s
\]
Now, dividing both sides by \( \sqrt{3} \):
\[
s = \frac{20}{\sqrt{3}} \approx 11.547 inches
\]

### Step 2: Perimeter Calculation
Since the perimeter \( P \) of an equilateral triangle is three times the side length:
\[
P = 3s = 3 \times 11.547 \approx 34.641 \text{ inches}
\]

So, the perimeter of the triangle is approximately **34.64 inches**.

Would you like further details or have any questions?

### Related Questions:
1. How do you calculate the area of this equilateral triangle?
2. What is the radius of the incircle in terms of the side length of the triangle?
3. How does the height of an equilateral triangle relate to its side length in general?
4. How do you calculate the perimeter of a regular polygon with an inscribed circle?
5. How can you calculate the side length of the triangle using trigonometric methods?

**Tip**: In an equilateral triangle, the relationship between the height and side length is a useful tool for many geometric problems.

This guide shows how to calculate the perimeter of an equilateral triangle with an inscribed circle. The height of the triangle is 10 inches, and we use key geometric formulas to find the side length and perimeter, resulting in approximately 34.64 inches. Suitable for students in Grades 9-12.

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