The diagram shows a seven-sided polygon, PQRSTUV. It is formed from two equilateral triangles PQW and STU of side-length 5 cm and 8 cm respectively. The two triangles overlap in an equilateral triangle of side-length 2 cm.

What is the perimeter of PQRSTUV?

To solve for the perimeter of the seven-sided polygon \( PQRSTUV \), we need to consider the given conditions and the information about the overlapping equilateral triangles.

### Step 1: Understanding the Overlapping Triangles
- **Triangle \( PQW \)** has a side length of \( 5 \) cm.
- **Triangle \( STU \)** has a side length of \( 8 \) cm.
- The two triangles overlap in an equilateral triangle \( X \) with a side length of \( 2 \) cm.

### Step 2: Perimeter of the Non-Overlapping Parts
Since the two triangles overlap, the overlapping part (side length \( 2 \) cm) is counted twice if we simply add up all sides of both triangles. Therefore, to find the perimeter of the polygon \( PQRSTUV \), we need to subtract the perimeter of the overlapping region.

### Step 3: Calculate Perimeter of Individual Triangles
- The perimeter of \( PQW \) is:
  \[
  \text{Perimeter of } PQW = 3 \times 5 = 15 \text{ cm}
  \]
- The perimeter of \( STU \) is:
  \[
  \text{Perimeter of } STU = 3 \times 8 = 24 \text{ cm}
  \]

### Step 4: Subtract the Overlap
The overlapping equilateral triangle \( X \) has a perimeter of:
\[
\text{Perimeter of overlap } = 3 \times 2 = 6 \text{ cm}
\]
Since this is counted twice (once in \( PQW \) and once in \( STU \)), we subtract it from the sum of the perimeters of \( PQW \) and \( STU \).

### Step 5: Total Perimeter of \( PQRSTUV \)
\[
\text{Perimeter of } PQRSTUV = (15 + 24 - 6) \text{ cm} = 33 \text{ cm}
\]

Thus, the perimeter of the polygon \( PQRSTUV \) is **33 cm**.

Would you like more details or have any questions? Here are some additional questions related to this problem:

1. How would the perimeter change if the overlap was a different size?
2. What would the perimeter be if the triangles didn't overlap?
3. How does the area of the polygon compare to the sum of the areas of the individual triangles?
4. Can the polygon \( PQRSTUV \) be inscribed in a circle? If so, what would be the radius?
5. What are the interior angles of the polygon \( PQRSTUV \)?

**Tip:** In problems involving overlapping shapes, always consider the overlapping regions separately to avoid counting them multiple times.

Learn how to calculate the perimeter of polygon PQRSTUV, which is formed by overlapping two equilateral triangles PQW and STU. Includes step-by-step solution, formulas for perimeter calculation, and details on handling overlapping regions. Suitable for high school students studying geometry and polygon properties.

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