Question Video: Determining Whether Two Polygons Are Similar | Nagwa Question Video: Determining Whether Two Polygons Are Similar | Nagwa

Question Video: Determining Whether Two Polygons Are Similar Mathematics • Second Year of Preparatory School

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Are the two polygons similar?

02:28

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.

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