Similarity of Polygons Practice Problems & Solutions

Master polygon similarity with step-by-step practice problems. Learn corresponding angles, proportional sides, and scale factors through interactive exercises.

📚Master Polygon Similarity Through Targeted Practice
  • Identify corresponding angles and sides in similar polygons
  • Calculate scale factors and similarity ratios between geometric figures
  • Solve for missing side lengths using proportional relationships
  • Determine if polygons are similar using angle and side criteria
  • Apply similarity concepts to rectangles, squares, and pentagons
  • Work with real-world applications of similar geometric figures

Understanding Similarity of Polygons

Complete explanation with examples

Similarity of Geometric Figures

Similarity in geometry refers to the relationship between two shapes that have the same shape but may differ in size. Two figures are similar if their corresponding angles are equal and the lengths of their corresponding sides are proportional. This means one figure can be obtained by resizing the other, either by scaling up or scaling down, without changing the shape.

While similarity is most commonly associated with triangles, it can apply to almost any shape or figure.

Similar geometric have these key properties:

  1. They have angles of the same size respectively. In other words, all corresponding angles between similar figures are equal, preserving the overall shape.
  2. Proportionality between the sides of such figures - the ratios of corresponding side lengths are the same across similar 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

Detailed explanation

Practice Similarity of Polygons

Test your knowledge with 5 quizzes

The two polygons below are similar.

Calculate the missing sides of the polygon ABCD.

333333333333555

Examples with solutions for Similarity of Polygons

Step-by-step solutions included
Exercise #1

Is rectangle ABCD similar to rectangle EFGH?

777333101010666AAABBBDDDCCCEEEFFFHHHGGG

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}

710≠12 \frac{7}{10}\ne\frac{1}{2}

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

Answer:

No

Video Solution
Exercise #2

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

141414XXX3.53.53.51.51.51.5

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

Video Solution
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?

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

Video Solution
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

Step-by-Step Solution

Answer:

Angle C = Angle O

Video Solution
Exercise #5

Which shapes are similar?

8888881212126668886661414146661414143.53.53.51.51.51.53.53.53.561.5

Step-by-Step Solution

Answer:

The rectangles are similar.

Video Solution

Frequently Asked Questions

Everything you need to know about Similarity of Polygons

What makes two polygons similar?

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Two polygons are similar when their corresponding angles are equal and their corresponding sides are proportional. This means one polygon can be obtained by scaling the other up or down without changing its shape.

How do you find the scale factor between similar polygons?

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To find the scale factor, divide the length of any side in the larger polygon by the corresponding side length in the smaller polygon. For example, if one side is 12 units and the corresponding side is 6 units, the scale factor is 12:6 or 2:1.

What is the difference between congruent and similar polygons?

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Congruent polygons have the same size and shape (scale factor of 1:1), while similar polygons have the same shape but may differ in size. Similar polygons include congruent polygons as a special case.

Can you have similar polygons with different orientations?

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Yes, similar polygons can be rotated, reflected, or translated and still remain similar. The key properties are equal corresponding angles and proportional corresponding sides, regardless of position or orientation.

How do you solve for missing sides in similar polygons?

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Set up a proportion using known corresponding sides, then cross-multiply to solve. If polygon A has sides 4 and x, and similar polygon B has corresponding sides 6 and 9, then 4/6 = x/9, so x = 6.

What are common examples of similar polygons in real life?

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Common examples include: photographs and their enlargements, maps at different scales, similar windows or doors in buildings, and geometric patterns in art and architecture where shapes are scaled proportionally.

Are all squares similar to each other?

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Yes, all squares are similar because they all have four right angles (equal corresponding angles) and all sides are equal within each square, making the ratios of corresponding sides always proportional.

What mistakes do students commonly make with polygon similarity?

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Common mistakes include: confusing similarity with congruence, incorrectly identifying corresponding sides, forgetting to check that ALL corresponding angles are equal, and setting up proportions incorrectly when solving for missing measurements.

Continue Your Math Journey

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