n 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
16
left parenthesis 1 comma 16 right parenthesis
2
22
left parenthesis 2 comma 22 right parenthesis
3
28
left parenthesis 3 comma 28 right parenthesis
(Simplify your answers.)
Part 2
Represent the relationship using words.
Multiply the number of triangles
in each figure by 6
,
to get the total length of the
top and bottom
sides of
the combined figure. Then add 10
for the total length of the
left and right
sides
of
the combined figure to get the entire perimeter.
Part 3
Represent the relationship using an equation.
yequals
enter your response here
(Simplify your answer. Type an expression using x as the variable.)
Part 1: Represent the relationship using a table.
| Number of Triangles, x | Perimeter, y | Ordered Pair (x,y) |
|------------------------|--------------|--------------------|
| 1 | 16 | (1, 16) |
| 2 | 22 | (2, 22) |
| 3 | 28 | (3, 28) |
From the given information and table, we can infer that each additional triangle increases the total perimeter by 6 units. To include this progression in the table, you might continue it as follows:
| Number of Triangles, x | Perimeter, y | Ordered Pair (x,y) |
|------------------------|--------------|--------------------|
| 4 | 34 | (4, 34) |
| 5 | 40 | (5, 40) |
| ... | ... | ... |
| n | y | (n, y) |
The perimeter of the figure when x is the number of triangles will increase by 6 each time an additional triangle is added. However, this is not the complete relationship, as there's an additional constant to consider.
Part 2: Represent the relationship using words.
The perimeter of the figure is determined by multiplying the number of triangles, x, by 6 to account for the top and bottom sides of the combined figure. To this product, we must add 10 to account for the static length of the left and right sides of the combined figure. So the perimeter is calculated by taking 6 times the number of triangles and then adding 10 for the edges that do not change in length with additional triangles.
Part 3: Represent the relationship using an equation.
The equation that represents this relationship, where y is the perimeter and x is the number of triangles, is as follows:
\(y = 6x + 10\)
This equation accounts for the 6 units added to the perimeter for each additional triangle (total length on the top and bottom sides) plus the constant 10 units for the left and right sides. This equation should satisfy the values given in the table for Part 1.