In the diagram below, what is the relationship between the number of triangles
and the perimeter of the figure they form? Represent this relationship using a table, words, an equation, and a graph. Let xequals
the
number of triangles
.
Let yequals
the
perimeter of the figure.
5655555666661 triangle2 triangles3 triangles
Question content area bottom
Part 1
Represent the relationship using a table. Complete the table below.
Number of Triangles
,
x
Perimeter, y
Ordered Pair (x,y)
1
3
left parenthesis 1 comma 3 right parenthesis
2
5
left parenthesis 2 comma 5 right parenthesis
3
7
left parenthesis 3 comma 7 right parenthesis
From the information provided, it appears that for every additional triangle added to the figure, the perimeter increases by 2 units. This is because each new triangle shares one side with the previous triangle, only adding one more outer side to the perimeter. The first triangle has a perimeter of 3 units, and for each additional triangle, we add 2 more units to the perimeter.
Part 1: Represent the relationship using a table.
| Number of Triangles, x | Perimeter, y | Ordered Pair (x,y) |
|------------------------|--------------|--------------------|
| 1 | 3 | (1, 3) |
| 2 | 5 | (2, 5) |
| 3 | 7 | (3, 7) |
| ... | ... | ... |
We can extend this table indefinitely, and for every new row—representing an additional triangle—the perimeter would increase by 2. The next few entries might look like:
| Number of Triangles, x | Perimeter, y | Ordered Pair (x,y) |
|------------------------|--------------|--------------------|
| 4 | 9 | (4, 9) |
| 5 | 11 | (5, 11) |
| 6 | 13 | (6, 13) |
We can describe the relationship in words as follows:
For each triangle we add to the figure, the perimeter increases by 2 units. This linear relationship can be represented by the equation y = 2x + 1, where x is the number of triangles, and y is the perimeter of the figure. This equation accounts for both the base perimeter of the first triangle and the addition of 2 units per each subsequent triangle.
Finally, let's represent this relationship using an equation:
y = 2x + 1
Where x is the number of triangles and y is the perimeter of the figure.
This can be graphed as a straight line, with x on the horizontal axis (number of triangles) and y on the vertical axis (perimeter). The graph will be a straight line that starts at (1, 3) and has a slope of 2. The graph will intercept the y-axis at y = 1 when x = 0, reflecting the y-intercept of the equation.
The points indicated in the table (1, 3), (2, 5), and (3, 7) will lie on this line, confirming the linear relationship between the number of triangles and the perimeter of the figure.