For each diagram find the relationship between the number of shapes and the perimeter of the figure they form. Represent this relationship using a table, words, an equation, and a graph.
Below there are 3 sets of Pentagon's there is one by itself with the number one on each of its sides next to it is two Pentagon's connected and next to that is three Pentagon's connected//
Thank you for your help in advance:)
Let's analyze each diagram and find the relationship between the number of shapes and the perimeter of the figure they form.
1) One Pentagon:
- Number of shapes: 1
- Perimeter: The perimeter of a pentagon will depend on the length of its sides. Let's assume the length of each side is 's'.
- Relationship: The perimeter of a single pentagon with side length 's' is 5s.
2) Two connected Pentagons:
- Number of shapes: 2
- Perimeter: When two pentagons are connected, two of their sides are shared. Therefore, the perimeter will depend on the length of the sides of each pentagon, as well as the shared sides. Let's assume the length of each side is 's' and the shared sides have a length of 'x'.
- Relationship: The perimeter of two connected pentagons with side length 's' and shared sides 'x' is 5s + 5s + x.
3) Three connected Pentagons:
- Number of shapes: 3
- Perimeter: Similarly, when three pentagons are connected, two sides of each adjacent pentagon are shared. Let's assume the length of each side is 's' and the shared sides have a length of 'x'.
- Relationship: The perimeter of three connected pentagons with side length 's' and shared sides 'x' is 5s + 5s + 5s + x + x.
Now let's represent this relationship using a table, words, an equation, and a graph:
Table:
Number of Shapes | Perimeter
-----------------|----------
1 | 5s
2 | 10s + x
3 | 15s + 2x
Words: The perimeter of a figure formed by one pentagon is equal to the length of its sides (5s). When two or three pentagons are connected, we add the lengths of their sides and any shared sides between them.
Equation: For any 'n' number of connected pentagons, the perimeter can be represented by the equation: Perimeter = 5 × n × s + (n-1) × x
Graph: Unfortunately, without knowing the specific lengths of the sides (s) and shared sides (x), we cannot accurately plot the graph as it would require numerical values.
To find the relationship between the number of shapes (pentagons) and the perimeter of the figure they form, we need to analyze each diagram and determine how the perimeter changes with the number of pentagons.
Diagram 1: One Pentagon
In this diagram, we have a single pentagon. The perimeter of the figure is simply the perimeter of one pentagon. Since a pentagon has five equal sides, we can calculate the perimeter by multiplying the length of one side by five. Let's say the length of one side of the pentagon is "s". Therefore, the equation for the perimeter of one pentagon is:
Perimeter = 5s
Diagram 2: Two Connected Pentagons
In this diagram, two pentagons are connected. To find the perimeter of the figure, we need to calculate the combined perimeter of both pentagons. Since the two pentagons share a side, we don't count that side twice. So, the perimeter of the figure will be the perimeter of one pentagon plus the perimeter of the other pentagon, minus the length of the common side. Hence, the equation for the perimeter of two connected pentagons is:
Perimeter = (5s) + (5s) - s = 9s
Diagram 3: Three Connected Pentagons
In this diagram, three pentagons are connected. Similarly, to find the perimeter of the figure, we calculate the combined perimeter of all three pentagons. However, this time we have two common sides. Therefore, the equation for the perimeter of three connected pentagons is:
Perimeter = (5s) + (5s) + (5s) - (2s) = 13s
Table:
Number of Pentagons (n) | Perimeter
----------------------------------
1 | 5s
2 | 9s
3 | 13s
Graph:
On the x-axis, we can represent the number of pentagons (n), and on the y-axis, we can represent the perimeter. Plotting the values from the table, we will see a linear relationship.
Equation:
The equation for the relationship between the number of pentagons (n) and the perimeter (P) is P = (4n + 1)s. The coefficient 4n represents the number of sides, and the constant 1 represents the additional side for the first pentagon. The variable s represents the length of each side.
In summary, the relationship between the number of shapes (pentagons) and the perimeter of the figure they form is linear, with the equation P = (4n + 1)s.