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.
