Question Video: Finding the Perimeter of a Polygon given the Perimeter of a Similar Polygon and the Lengths of Two Corresponding Sides Mathematics

Video Transcript

Given that 𝐴𝐵𝐶𝐷 is similar to 𝑋𝑌𝑍𝐿 and the perimeter of 𝐴𝐵𝐶𝐷 is 66.2 centimeters, calculate the perimeter of 𝑋𝑌𝑍𝐿, giving your answer to two decimal places.

We’re told that polygons 𝐴𝐵𝐶𝐷 and 𝑋𝑌𝑍𝐿 are similar, which means that two key things are true. Firstly corresponding angles are congruent, and secondly corresponding sides are proportional. Here, we’re asked about the perimeter of 𝑋𝑌𝑍𝐿. So this second property is going to be useful.

As the perimeter is simply the sum of the side lengths, the perimeters of similar polygons are in the same proportion as the corresponding side lengths themselves. As we’ve been given the perimeter of 𝐴𝐵𝐶𝐷, we can calculate the perimeter of 𝑋𝑌𝑍𝐿 if we know what this proportion is. To work this out, let’s see if we can identify a pair of corresponding sides on the two polygons whose lengths we know.

The ordering of the letters in the similarity statement is important, because it reveals which vertices in the two shapes are corresponding. Vertex 𝐴 corresponds with vertex 𝑋, vertex 𝐵 corresponds with vertex 𝑌, and so on. We can identify that side 𝐵𝐶 is corresponding with side 𝑌𝑍. And we’ve been given the lengths of both of these sides. 𝐵𝐶 is 16 centimeters, and 𝑌𝑍 is 10 centimeters.

We can then write down the proportion. And as it’s the perimeter of 𝑋𝑌𝑍𝐿 that we want to calculate later, we’ll use lengths from this polygon in the numerator of the fraction. We have 𝑌𝑍 over 𝐵𝐶 is equal to 10 over 16. This fraction can be simplified to five over eight.

We can now form an equation using the perimeters of the two polygons. As they’re in the same proportion as the individual side lengths, we have that the perimeter of 𝑋𝑌𝑍𝐿 over the perimeter of 𝐴𝐵𝐶𝐷 is equal to five over eight. We can then substitute 66.2 for the perimeter of 𝐴𝐵𝐶𝐷. To solve this equation, we multiply both sides by 66.2 and then evaluate. This gives 331 over eight, which is 41.375.

The question specifies that we should give our answer to two decimal places. So, rounding as required and including the units, we’ve found that the perimeter of polygon 𝑋𝑌𝑍𝐿 is 41.38 centimeters to two decimal places.