Similarity of Geometric Figures

🏆Practice similarity of polygons

The similarity between geometric figures is met when they have angles of the same size respectively and there is also proportionality between the sides of such figures. 

In an intuitive way, just as it happens with triangles, two similar figures are, in fact, an enlargement of the other.

Similarity of geometric figures image

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Test yourself on similarity of polygons!

einstein

Is rectangle ABCD similar to rectangle EFGH?

777333101010666AAABBBDDDCCCEEEFFFHHHGGG

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Similarity of Geometric Figures

Example 1

Let's demonstrate this topic with an example.

We have an illustration of two similar rectangles, ABCD ABCD and KLMN KLMN .

two similar Rectangles

In both rectangles all angles are right angles (equivalent to 90º 90º ).

Moreover, each side of the large rectangle KLMN KLMN is greater than the respective side in the small rectangle ABCD ABCD .

That is, KL=12 KL=12 in the large rectangle KLMN KLMN is twice as long as AB=6 AB=6 in the small rectangle ABCD ABCD , and KN=8 KN=8 in the large rectangle KLMN KLMN is twice as long as AB=4 AB=4 in the small rectangle ABCD ABCD .


Example 2

These two squares are similar:

Similarity of geometric figures image

The two corresponding angles are equal since all angles are right angles. The ratio between the corresponding sides, that is, the scale factor is
2:1 2:1

or, in other words, each side of the larger square measures twice as much as each side of the small square


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Example 3 - Similar Figures

The two pentagons in the illustration are similar, meaning, the corresponding angles are equal. The ratio of similarity is 

EFAB=32=1.51 \frac{EF}{AB}=\frac{3}{2}=\frac{1.5}{1}

The two pentagons in the illustration are similar

That is, the length of each side in the pentagon FGHIJ FGHIJ is 1.5 1.5 times greater than that of its corresponding side in the pentagon ABCDE ABCDE


If you are interested in this article, you might also be interested in the following articles

  • Similarity of triangles and polygons
  • Ratio of similarity
  • Similar triangles
  • Criterion of similarity between two triangles

In the blog of Tutorela you will find a wide variety of mathematics articles


Examples and exercises with solutions on similarity of geometric figures

Exercise #1

Is rectangle ABCD similar to rectangle EFGH?

777333101010666AAABBBDDDCCCEEEFFFHHHGGG

Video Solution

Step-by-Step Solution

We first need to verify the ratio of similarity.

We examine if:

ABEF=ACEG \frac{AB}{EF}=\frac{AC}{EG}

To do this, we substitute our values in:

710=36 \frac{7}{10}=\frac{3}{6}

71012 \frac{7}{10}\ne\frac{1}{2}

The ratio is not equal, therefore the rectangles are not similar.

Answer

No

Exercise #2

Look at the two similar rectangles below and calculate the perimeter of the larger rectangle.

141414XXX3.53.53.51.51.51.5

Video Solution

Step-by-Step Solution

Let's remember that in a rectangle there are two pairs of parallel and equal sides.

We will call the small triangle 1 and the large triangle 2.

We calculate the perimeter of the small triangle:

P1=2×3.5+2×1.5=10 P_1=2\times3.5+2\times1.5=10 Since we know that the rectangles are similar:

3.514=p1p2 \frac{3.5}{14}=\frac{p_1}{p_2}

We place the data we know for the perimeter:

3.514=10p2 \frac{3.5}{14}=\frac{10}{p_2}

3.514×p2=10 \frac{3.5}{14}\times p_{_2}=10

p2=10×143.5 p_2=10\times\frac{14}{3.5}

P2=40 P_2=40

Answer

40 cm

Exercise #3

1027.51.5The two parallelograms above are similar. The ratio between their sides is 3:4.

What is the ratio between the the areas of the parallelograms?

Video Solution

Step-by-Step Solution

The square of the ratio between the sides is equal to the ratio between the areas of the parallelograms:

32:42=9:16 3^2:4^2=9:16

Answer

9:16

Exercise #4

In front of you are two hexagons with a similarity ratio. Which angles are corresponding?

4.54.54.50.750.750.753334443336663330.50.50.52222.662.662.66222444AAABBBCCCDDDEEEFFFMMMNNNOOOPPPRRRJJJ

Video Solution

Answer

Angle C = Angle O

Exercise #5

Which shapes are similar?

8888881212126668886661414146661414143.53.53.51.51.51.53.53.53.561.5

Video Solution

Answer

The rectangles are similar.

Do you know what the answer is?
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