In the diagram below, what is the relationship between the number of parallelograms, n, and the perimeter, P, of the figure they form?
The image shows three figures composed of congruent parallelograms.
The first figure is composed of 1 parallelogram. The left and right sides are each labeled 4 and the top and bottom sides are each labeled 5.
The second figure is composed of 2 parallelograms that share a side, so that the left and right sides of the figure are each labeled 4, and the top and bottom sides are each composed of 2 segments labeled 5.
The third figure is composed of 3 parallelograms so that 2 adajcent parallelograms share 1 side. The left and and right sides of the figure are each labeled 4, and the top and bottom sides are each composed of 3 segments labeled 5.
(1 point)
A) The equation for the perimeter is P = 10n + 4.
B) The equation for the perimeter is P = 4n + 10.
C) The equation for the perimeter is P = 4n + 5.
D) The equation for the perimeter is P = 10n + 8.
The correct answer is A) The equation for the perimeter is P = 10n + 4.
In each figure, the left and right sides of the parallelograms are each labeled 4. Since there are n parallelograms in the figure, the total length of the left and right sides is 4n.
The top and bottom sides of the figures are composed of segments labeled 5. In the first figure, there is 1 segment, in the second figure, there are 2 segments, and in the third figure, there are 3 segments. So, the total length of the top and bottom sides can be expressed as 5*(1 + 2 + 3) = 5*(n(n+1)/2).
Therefore, the perimeter, P, is given by P = 4n + 5*(n(n+1)/2).
Simplifying this expression, we get P = 4n + 5n(n+1)/2.
Further simplifying, we get P = 4n + (5n^2 + 5n)/2.
Expanding this expression, we get P = 4n + (5n^2 + 5n)/2 = 4n + (10n^2 + 10n)/2.
Finally, simplifying again, we get P = 4n + 10n^2 + 10n = 10n + 10n^2 + 4.
Therefore, the equation for the perimeter is P = 10n + 4.