The task asks to determine which of the polygons has the greatest perimeter. The vertices of the three polygons are provided as coordinates. Let's break this down:

• Polygon A (ABCDE):

  • $A(2, 7), B(2, 4), C(4, 2), D(6, 4), E(4, 7)$

• Polygon B (EFG):

  • $E(-2, 2), F(1, 3), G(4, 1)$

• Polygon C (HIJK):

  • $H(2, 3), I(6, 3), J(9, -5), K(6, -1)$

To determine the perimeters, I will calculate the distance between consecutive vertices for each polygon using the distance formula:

$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

I will now proceed to calculate the perimeters of each polygon.The perimeters of the polygons are as follows:

• Polygon A (ABCDE): $14.26$ units
• Polygon B (EFG): $12.85$ units
• Polygon C (HIJK): $23.20$ units

Therefore, Polygon C (HIJK) has the greatest perimeter.

Would you like further details or explanations? Here are five related questions you might explore:

1. How do you calculate the area of each polygon?
2. Can the shape of a polygon influence its perimeter with the same set of vertices?
3. How does the distance formula work in different dimensions?
4. What is the role of coordinates in geometry when determining lengths and perimeters?
5. How would adding an extra vertex affect the perimeter of a polygon?

Tip: The perimeter of a polygon is the total length around the figure, and it increases as the distances between vertices increase.