Geometry Worksheets | Similarity Worksheets - Free Printable

Let's solve each problem step by step using the concept of similar polygons. When two polygons are similar, their corresponding sides are proportional, and the ratio of their corresponding sides is called the scale factor.

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Problem 1

Given:
- Left polygon has a side of 45.
- Right polygon has a corresponding side of 60.
- The other side on the right is 8.
- We need to find the missing side (marked with ?) on the left.

We can set up a proportion based on the known sides:

$$
\frac{?}{8} = \frac{45}{60}
$$

Simplify:
$$
\frac{?}{8} = \frac{3}{4}
$$

Multiply both sides by 8:
$$
? = 8 \times \frac{3}{4} = 6
$$

✔ Answer: 6

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Problem 2

Given:
- Scale factor from left to right is 3:5
- Right rectangle has a side of 75
- Find the missing side on the left

Let the missing side be $ x $. Since scale factor is left : right = 3 : 5, then:

$$
\frac{x}{75} = \frac{3}{5}
$$

Cross-multiply:
$$
5x = 225 \Rightarrow x = 45
$$

✔ Answer: 45

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Problem 3

Given:
- Scale factor from left to right is 1:3
- Right trapezoid has a base of 30
- Find the missing base on the left

Let $ x $ be the missing side on the left.

$$
\frac{x}{30} = \frac{1}{3}
\Rightarrow x = \frac{30}{3} = 10
$$

✔ Answer: 10

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Problem 4

Given:
- Scale factor from left to right is 5:6
- Left polygon has a side of 20
- Find the missing side on the right

Let $ x $ be the missing side on the right.

$$
\frac{20}{x} = \frac{5}{6}
\Rightarrow 5x = 120 \Rightarrow x = 24
$$

✔ Answer: 24

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Problem 5

Given:
- Scale factor from left to right is 1:4
- Left triangle has a side of 2
- Find $ x $, the corresponding side on the right

$$
\frac{2}{x} = \frac{1}{4}
\Rightarrow x = 8
$$

✔ Answer: 8

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Problem 6

Given:
- Scale factor from left to right is 1:6
- Left parallelogram has a side of 3
- Find $ x $, the corresponding side on the right

$$
\frac{3}{x} = \frac{1}{6}
\Rightarrow x = 18
$$

✔ Answer: 18

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Problem 7

Given:
- Two rectangles are similar
- Left rectangle: 10 (height), 24 (width)
- Right rectangle: 36 (height), $ x $ (width)

Since they're similar, the ratios of corresponding sides must be equal.

Set up proportion:
$$
\frac{10}{36} = \frac{24}{x}
$$

Cross-multiply:
$$
10x = 864 \Rightarrow x = 86.4
$$

Alternatively, simplify:
$$
\frac{10}{36} = \frac{5}{18}, \quad \text{so } \frac{5}{18} = \frac{24}{x}
\Rightarrow 5x = 432 \Rightarrow x = 86.4
$$

✔ Answer: 86.4

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Problem 8

Given:
- Two quadrilaterals are similar
- Left: one side is $ x $, another is 7
- Right: corresponding sides are 24 and 14

We need to determine which sides correspond. Since the figures are similar, we assume the sides are in order.

So, the side of length 7 corresponds to 14, and $ x $ corresponds to 24.

So:
$$
\frac{x}{24} = \frac{7}{14} = \frac{1}{2}
\Rightarrow x = 12
$$

✔ Answer: 12

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✔ Final Answers:

1) 6
2) 45
3) 10
4) 24
5) 8
6) 18
7) 86.4
8) 12

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🔍 Explanation Summary:

- In similar polygons, corresponding sides are proportional.
- Use the scale factor or set up a proportion between corresponding sides.
- Always make sure you match the correct sides (based on position or labeling).
- If scale factor is given as "Left to Right", use it directly:
$$
\frac{\text{Left}}{\text{Right}} = \text{Scale Factor}
$$

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