What is the total perimeter of this shape? 

To find the total perimeter of the combined shape (the green square and the purple triangle), we'll need to calculate the perimeter of each part and then sum them.

### 1. **Perimeter of the Green Square:**
   - The side length of the square is \(4\) units (since the coordinates are \((4,4)\) to \((8,4)\)).
   - Perimeter of the square = \(4 \times \text{side length} = 4 \times 4 = 16\) units.

### 2. **Perimeter of the Purple Triangle:**
   - The base of the triangle lies along the top of the square, from \((5,8)\) to \((7,8)\). So the base length is \(2\) units.
   - The height is the vertical distance from the top of the square to the highest point of the triangle, from \((6,10)\) to \((6,8)\), so the height is \(2\) units.
   
   - The hypotenuse (the side opposite the right angle) can be found using the Pythagorean theorem:
     \[
     \text{Hypotenuse} = \sqrt{(\text{base})^2 + (\text{height})^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \text{ units}
     \]

   - The perimeter of the triangle is the sum of the base, height, and hypotenuse:
     \[
     \text{Perimeter of the triangle} = 2 + 2 + 2\sqrt{2} \approx 2 + 2 + 2 \times 1.414 \approx 6.828 \text{ units}
     \]

### 3. **Total Perimeter of the Combined Shape:**
   - The shared side between the square and the triangle should only be counted once, so the total perimeter is:
     \[
     \text{Total perimeter} = \text{Perimeter of the square} + \text{Perimeter of the triangle} - \text{base of the triangle}
     \]
     \[
     \text{Total perimeter} = 16 + 6.828 - 2 \approx 20.828 \text{ units}
     \]

The total perimeter of the combined shape is approximately **20.828 units**.

Would you like any further details or clarifications?

### Related Questions:
1. How can we accurately calculate the hypotenuse of a right triangle on a coordinate plane?
2. What is the difference between the perimeter and the area of a shape?
3. How does the perimeter of the combined shape change if the triangle's height is increased?
4. Can the perimeter be calculated differently if the triangle is repositioned?
5. How would the perimeter change if the square was larger?
6. How does the coordinate system help in calculating perimeters and areas?
7. What is the significance of the Pythagorean theorem in perimeter calculations?
8. How can we check the accuracy of our perimeter calculations?

### Tip:
When calculating perimeters of combined shapes, remember to account for any shared sides so that you don’t double-count them.

Learn how to calculate the total perimeter of a combined shape consisting of a green square and a purple triangle. Understand the step-by-step process using geometric principles and the Pythagorean theorem. Suitable for Grades 7-9 students.

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