https://r2.erweima.ai/mathbot/image/1726721626079knerg2kr.jpg

The image shows a triangle with vertices at points \( A(-4, 2) \), \( B(6, 2) \), and \( C(6, 12) \). The question seems to ask for the polygon type and its perimeter.

### Polygon Type:
This is a **right triangle**. The right angle is formed at point \( B \) because the sides \( \overline{AB} \) and \( \overline{BC} \) are perpendicular (one is horizontal, and the other is vertical).

### Perimeter:
To calculate the perimeter, we need the lengths of the sides \( \overline{AB} \), \( \overline{BC} \), and \( \overline{AC} \).

- \( \overline{AB} \) is a horizontal segment, so its length is the difference in the x-coordinates:
  \[
  \overline{AB} = |6 - (-4)| = 10
  \]
  
- \( \overline{BC} \) is a vertical segment, so its length is the difference in the y-coordinates:
  \[
  \overline{BC} = |12 - 2| = 10
  \]

- \( \overline{AC} \) is the hypotenuse of the right triangle, and we can calculate it using the Pythagorean theorem:
  \[
  \overline{AC} = \sqrt{(6 - (-4))^2 + (12 - 2)^2} = \sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2} \approx 14.14
  \]

### Perimeter:
Now, the perimeter \( P \) is the sum of all three sides:
\[
P = \overline{AB} + \overline{BC} + \overline{AC} = 10 + 10 + 14.14 \approx 34.14
\]

### Final Answer:
- **Polygon:** Right triangle
- **Perimeter:** \( 34.14 \) units (approximately)

Let me know if you'd like more details or have further questions.

Here are five related questions to consider:
1. How do you calculate the area of this right triangle?
2. Can you verify the length of the hypotenuse using another method?
3. What is the distance between any two given points on a coordinate plane?
4. How would the perimeter change if the triangle was not a right triangle?
5. How does the Pythagorean theorem help in geometry problems involving right triangles?

**Tip:** Always double-check your coordinate points before starting distance calculations!

Identify the polygon and calculate the perimeter based on the triangle with vertices A(-4, 2), B(6, 2), and C(6, 12).

Explore how to identify a right triangle on a coordinate plane and calculate its perimeter. Using points A(-4, 2), B(6, 2), and C(6, 12), this problem demonstrates the application of the distance formula and Pythagorean theorem. Suitable for students in Grades 7-10.

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