Video Transcript
Are the two polygons similar?
Let’s begin by recalling what we
mean by the word “similar.” We say that two polygons are
similar if their corresponding angles are congruent and their corresponding sides
are in proportion.
So let’s begin by looking at the
sides. The side 𝐵𝐶 in quadrilateral
𝐴𝐵𝐶𝐷 is corresponding to side 𝐹𝐺 in quadrilateral 𝐸𝐹𝐺𝐻. We can write the ratio of sides
𝐵𝐶 over 𝐹𝐺 as 28 over 14, which simplifies to two. If these quadrilaterals are
similar, then as we can see in the definition above, all the other corresponding
pairs of sides would have to be in this same proportion of two.
So let’s compare the sizes of
another pair of corresponding sides, for which we are given the lengths of. 𝐴𝐵 is 31 centimeters and 𝐸𝐹 is
23 centimeters. Writing these the same way round as
before, that is, with the side from 𝐴𝐵𝐶𝐷 as the numerator and the side on
𝐸𝐹𝐺𝐻 as the denominator, we would put 𝐴𝐵 over 𝐸𝐹, which gives us 31 over
23. But notice that this fraction
doesn’t simplify any further. And most importantly here, it is
definitely not equal to two. That means that the sides in
𝐴𝐵𝐶𝐷 and 𝐸𝐹𝐺𝐻 are not in proportion. And therefore, we can answer the
question “Are the two polygons similar?” as no.
When we are demonstrating that two
shapes are not similar, it is enough to show that a pair of sides are not in
proportion as we did here. Or we might demonstrate that a pair
of corresponding angles are not equal. However, as we can see in the
definition above, to prove two shapes are similar, we do need to prove that every
pair of corresponding angles are congruent and every pair of corresponding sides are
in proportion.
