Think & Click: Geometric Induction
Now, you try. Complete the Think & Click activity by looking at the problem below, thinking about it, and then clicking on it to reveal the solution.
Each of the following figures is a regular pentagon. A regular pentagon is a polygon formed by five equal sides and five equal angles. That is, to find the perimeter of a regular pentagon we must multiply a side length by 5.
The perimeters of all these pentagons follow a general pattern. The perimeters of A, B, C, and D are 15, 25, 35, and 45, respectively.
Find the perimeter of the regular pentagon E.
Let’s observe the common pattern among the given pentagons. Denote the perimeters of A, B, C, D, and E by PA PB, PC, PD, and PE. Then
PA = 15 = 5 × 3
PB = 25 = 5 × 5
PC = 35 = 5 × 7
PD = 45 = 5 × 9
These products show that each side of the first pentagon measures 3. Then the sides of each pentagon thereafter are increased by 2 with respect to the previous one. Applying this pattern, we conclude that a side of E measures 9 + 2 = 11. Then, the perimeter of E is 5 × 11 = 55
