Video Transcript
A polygon has sides of length two
centimeters, four centimeters, three centimeters, eight centimeters, and four
centimeters. A second similar polygon has a
perimeter of 31.5 centimeters. What are the lengths of its
sides?
Let’s begin by recalling that
similar polygons have corresponding angles congruent and corresponding sides in
proportion. Here we are given the five side
lengths of this polygon, which must be a pentagon, and we need to work out the side
lengths of the second polygon using only the information about its perimeter. Because the side lengths of similar
polygons are in proportion, then the perimeter, which is also a measure of length,
will be in the same proportion. We can calculate the scale factor
from the first polygon to the second polygon by working out the perimeter of the
second polygon divided by the perimeter of the first polygon. We are given that the perimeter of
the second polygon is 31.5 centimeters.
And we can calculate the perimeter
of the first polygon by adding the side lengths. Two plus four plus three plus eight
plus four centimeters will give us a perimeter of 21 centimeters. The scale factor is therefore 31.5
over 21, which can be simplified to three over two. So now to find the side length in
the second polygon, we multiply each corresponding side length in the first polygon
by the scale factor of three over two. The first side length will
therefore be calculated as two times three over two. This will be equal to three
centimeters. Next, four centimeters times three
over two is six centimeters. The third side is three centimeters
times three over two, which is 4.5 centimeters. The fourth and fifth sides can be
calculated as 12 centimeters and six centimeters.
As a useful check of our answer, we
know that the perimeter of the second polygon should be 31.5. And when we add three centimeters,
six centimeters, 4.5 centimeters, 12 centimeters, and six centimeters, we do indeed
get a perimeter of 31.5 centimeters, which confirms our answer for the five side
lengths.
