In the diagram​ below, what is the relationship between the number of pentagons and the perimeter of the figure they​ form? Represent this relationship using a​ table, words, an​ equation, and a graph. Let xequalsthe number of pentagons. Let yequalsthe perimeter of the figure.

Question

3
3
3
3
1 pentagon
2 pentagons
3 pentagons
3
Question content area bottom
Part 1
Represent the relationship using a table. Complete the table below.
Number of Pentagons​, x
​Perimeter, y
Ordered Pair​ (x,y)
1
  
15
  
left parenthesis 1 comma 15 right parenthesis
2
  
24
  
left parenthesis 2 comma 24 right parenthesis
3
  
33
  
left parenthesis 3 comma 33 right parenthesis
​(Simplify your​ answers.)
Part 2
Represent the relationship using words.
Multiply the number of pentagons in each figure by
  
enter your response here​, since all pentagons will have at least
▼
two
three
sides that contribute to the perimeter of the combined figure. Then add
  
enter your response here for the
▼
two
three
unaccounted for sides on the left and right ends of the combined figure to get the entire perimeter.

Answer 1

Part 1: Represent the Relationship Using a Table

Here’s the completed table showing the relationship between the number of pentagons and the perimeter of the figure they form.

| Number of Pentagons (x) | Perimeter (y) | Ordered Pair (x,y) | |--------------------------|----------------|------------------------------| | 1 | 15 | (1, 15) | | 2 | 24 | (2, 24) | | 3 | 33 | (3, 33) |

Part 2: Represent the Relationship Using Words

Multiply the number of pentagons in each figure by 9, since all pentagons will have at least 3 sides that contribute to the perimeter of the combined figure. Then add 6 for the 3 unaccounted-for sides on the left and right ends of the combined figure to get the entire perimeter.

Relationship Summary

• Equation: The perimeter \( y \) can be expressed as: \[ y = 9x + 6 \] where \( x \) is the number of pentagons.

Part 3: Graph the Relationship

To graph the relationship, plot the ordered pairs from the table on a Cartesian coordinate system. Here's a description of how the graph would look:

1. The x-axis represents the number of pentagons (x), and the y-axis represents the perimeter (y).
2. Plot the points:
   • (1, 15)
   • (2, 24)
   • (3, 33)
3. Draw a line through these points; since this is a linear relationship, it will form a straight line.

The line will start near the y-axis at y=15 when x=1 and will have a slope of 9, reflecting the increase in perimeter with each additional pentagon.

Final Notes

• The relationship is linear, indicating that as the number of pentagons increases, the perimeter increases at a constant rate.
• Each pentagon contributes consistently to the overall perimeter based on its geometrical properties.

Feel free to ask if you need more details or further explanation on any part!