https://file.aiquickdraw.com/mathbot/file/1729556094421if2ako1c.jpg

Let's break down this problem. The figure consists of part of a circle and part of a square, and we are tasked with finding the perimeter of the figure.

### Steps to find the perimeter:
1. **Side lengths of the square:**
   - The square has a side of 5 units, as given in the image.
   
2. **Quarter-circle:**
   - The figure includes a quarter-circle attached to the square. The radius of this quarter-circle is also 5 units (since it's attached to the side of the square).

3. **Perimeter Calculation:**
   - For the square portion, we include two sides: one vertical side (length 10) and the horizontal side of the square (5 units).
   - The quarter-circle adds the arc length to the perimeter.
   
4. **Arc length of quarter-circle:**
   The arc length of a circle is given by the formula:
   \[
   \text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r
   \]
   Since this is a quarter-circle, \(\theta = 90^\circ\), and the radius \(r = 5\):
   \[
   \text{Arc Length} = \frac{90}{360} \times 2\pi \times 5 = \frac{1}{4} \times 10\pi = 2.5\pi
   \]
   Approximate value: \(2.5\pi \approx 7.85\) units.

### Total Perimeter:
- Vertical side = 10 units
- Horizontal side of the square = 5 units
- Quarter-circle arc length = \(7.85\) units

Adding them up:
\[
\text{Total Perimeter} = 10 + 5 + 7.85 = 22.85 \text{ units}
\]

So, the perimeter of the figure is approximately **22.85 units**.

Do you have any questions or need further details on this solution?

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Here are 5 related questions:
1. How would the area of this figure be calculated?
2. What is the perimeter if the radius of the quarter-circle was doubled?
3. How would the perimeter change if the figure was a semicircle instead of a quarter-circle?
4. Can you derive the general formula for the perimeter of a figure made of a square and part of a circle?
5. What would the total perimeter be if the square side length was different from the circle’s radius?

**Tip**: Always ensure you correctly identify which portions of a figure contribute to the perimeter versus the area to avoid confusion.

What is the perimeter of a figure made of part of a circle and part of a square, where the square's side is 5 units and the figure includes a quarter-circle with radius 5 units?

Learn how to find the perimeter of a figure made from a square and a quarter-circle with a detailed solution. This problem covers key geometry concepts, including perimeter calculation and arc length of a circle. Suitable for Grades 7-9 students.

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