To find the perimeter of the quadrilateral with the given vertices, we'll calculate the distance between each pair of adjacent points, then sum these distances.

The vertices are:

• $A(-2, 1)$
• $B(3, 1)$
• $C(3, -7)$
• $D(-2, -7)$

Step 1: Calculate the distances between adjacent points.

1. Distance $AB$: $AB = \sqrt{(3 - (-2))^2 + (1 - 1)^2} = \sqrt{(3 + 2)^2 + 0^2} = \sqrt{5^2} = 5$

2. Distance $BC$: $BC = \sqrt{(3 - 3)^2 + (-7 - 1)^2} = \sqrt{0^2 + (-8)^2} = \sqrt{64} = 8$

3. Distance $CD$: $CD = \sqrt{(-2 - 3)^2 + (-7 - (-7))^2} = \sqrt{(-5)^2 + 0^2} = \sqrt{25} = 5$

4. Distance $DA$: $DA = \sqrt{(-2 - (-2))^2 + (1 - (-7))^2} = \sqrt{0^2 + 8^2} = \sqrt{64} = 8$

Step 2: Calculate the perimeter.

The perimeter $P$ is the sum of the distances: $P = AB + BC + CD + DA = 5 + 8 + 5 + 8 = 26$

Final Answer:

The perimeter of the quadrilateral is $\boxed{26}$.

Would you like more details on any part of the solution? Here are five related questions to deepen your understanding:

1. How do you calculate the distance between any two points in a coordinate plane?
2. What is the difference between the perimeter and the area of a shape?
3. How would the perimeter change if the coordinates of one point were different?
4. Can the perimeter formula be generalized for any quadrilateral in a coordinate plane?
5. How would you find the perimeter if the quadrilateral were not axis-aligned?

Tip: Always ensure you have the correct order of vertices when calculating the perimeter of a polygon to avoid missing sides or duplicating them.