If the pentagon 𝐴𝐵𝐶𝐷𝐸 is similar to the pentagon 𝑃𝑄𝑅𝑆𝑇. Find the scale factor of 𝐴𝐵𝐶𝐷𝐸 to 𝑃𝑄𝑅𝑆𝑇 and the perimeter of 𝑃𝑄𝑅𝑆𝑇.
In order to calculate the scale factor, we need to find corresponding sides of the two pentagons. In this case, 𝐶𝐷 is equal to 21 and 𝑃𝑇 is equal to 14. This means that the scale factor will be 21 divided by 14. This can be simplified to three over two or 1.5 as a decimal. This means that the larger pentagon has length 1.5 times as long as the smaller pentagon.
The perimeter of any shape is the total distance around the outside of the shape. In this example, we can work out the perimeter of pentagon 𝐴𝐵𝐶𝐷𝐸 by adding 24, 24, 21, 21, and 27. This is because length 𝐵𝐶 is the same as length 𝐴𝐵. In the same way, length 𝐶𝐷 is the same as length 𝐴𝐸. Adding these numbers gives us the perimeter of the larger pentagon of 117.
To calculate the perimeter of the smaller pentagon, we need to divide this answer 117 by the scale factor — in this case 1.5 or three over two. 117 divided by 1.5 is equal to 78. This means that the perimeter of the smaller pentagon is 78.
We could also have worked out the perimeter of the smaller pentagon by working out each individual length — the length of 𝑃𝑄, 𝑄𝑅, 𝑅𝑆, and 𝑆𝑇. This could be done by dividing the corresponding length in the larger pentagon by 1.5. Adding these numbers — 16, 16, 14, 18, and 14 — also gives us an answer of 78.