Side Lengths in Similar Figures

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Key Ideas Easy Medium Hard

What to Remember

Same ratio

Figures are similar when every pair of matching sides has the same ratio.

Checking one pair is not enough. If two pairs have different ratios, the figures are not similar.

A missing side

Put one pair of matching sides in a proportion, then the other pair.

The unknown side is the known side of its pair multiplied by the scale factor from the first pair.

Scale factor 2 or 3

A scale factor of 2 or 3 gives a whole-number side when you start from a whole number.

A fraction scale factor

A scale factor can be one half, three halves, or five halves and still give a clean length.

Level 1

Easy

The scale factor is 2 or 3.

Example

The figures are similar. A pair of matching sides is 4 and 8. Another side of the smaller figure is 5. Find the matching side x.

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

48 = 5x

The scale factor from the smaller side to the matching larger side is 2.

x = 5 × 2

x = 10

Answer: 10

The figures are similar. A pair of matching sides is 3 and 6. Another side of the smaller figure is 4. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

36 = 4x

The scale factor from the smaller side to the matching larger side is 2.

x = 4 × 2

x = 8

Answer: 8

The figures are similar. A pair of matching sides is 5 and 10. Another side of the smaller figure is 2. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

510 = 2x

The scale factor from the smaller side to the matching larger side is 2.

x = 2 × 2

x = 4

Answer: 4

The figures are similar. A pair of matching sides is 2 and 4. Another side of the smaller figure is 7. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

24 = 7x

The scale factor from the smaller side to the matching larger side is 2.

x = 7 × 2

x = 14

Answer: 14

The figures are similar. A pair of matching sides is 6 and 12. Another side of the smaller figure is 3. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

612 = 3x

The scale factor from the smaller side to the matching larger side is 2.

x = 3 × 2

x = 6

Answer: 6

The figures are similar. A pair of matching sides is 4 and 8. Another side of the smaller figure is 6. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

48 = 6x

The scale factor from the smaller side to the matching larger side is 2.

x = 6 × 2

x = 12

Answer: 12

The figures are similar. A pair of matching sides is 2 and 6. Another side of the smaller figure is 5. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

26 = 5x

The scale factor from the smaller side to the matching larger side is 3.

x = 5 × 3

x = 15

Answer: 15

The figures are similar. A pair of matching sides is 3 and 9. Another side of the smaller figure is 4. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

39 = 4x

The scale factor from the smaller side to the matching larger side is 3.

x = 4 × 3

x = 12

Answer: 12

The figures are similar. A pair of matching sides is 5 and 15. Another side of the smaller figure is 2. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

515 = 2x

The scale factor from the smaller side to the matching larger side is 3.

x = 2 × 3

x = 6

Answer: 6

The figures are similar. A pair of matching sides is 4 and 12. Another side of the smaller figure is 3. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

412 = 3x

The scale factor from the smaller side to the matching larger side is 3.

x = 3 × 3

x = 9

Answer: 9

The figures are similar. A pair of matching sides is 1 and 3. Another side of the smaller figure is 6. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

13 = 6x

The scale factor from the smaller side to the matching larger side is 3.

x = 6 × 3

x = 18

Answer: 18

Level 2

Medium

The scale factor is one half, three halves, or five halves.

Example

The figures are similar. A pair of matching sides is 4 and 6. Another side of the smaller figure is 6. Find the matching side x.

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

46 = 6x

The scale factor from the smaller side to the matching larger side is 32.

x = 6 × 32

x = 9

Answer: 9

The figures are similar. A pair of matching sides is 8 and 4. Another side of the smaller figure is 6. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

84 = 6x

The scale factor from the smaller side to the matching larger side is 12.

x = 6 × 12

x = 3

Answer: 3

The figures are similar. A pair of matching sides is 10 and 5. Another side of the smaller figure is 4. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

105 = 4x

The scale factor from the smaller side to the matching larger side is 12.

x = 4 × 12

x = 2

Answer: 2

The figures are similar. A pair of matching sides is 6 and 3. Another side of the smaller figure is 10. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

63 = 10x

The scale factor from the smaller side to the matching larger side is 12.

x = 10 × 12

x = 5

Answer: 5

The figures are similar. A pair of matching sides is 2 and 3. Another side of the smaller figure is 4. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

23 = 4x

The scale factor from the smaller side to the matching larger side is 32.

x = 4 × 32

x = 6

Answer: 6

The figures are similar. A pair of matching sides is 6 and 9. Another side of the smaller figure is 2. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

69 = 2x

The scale factor from the smaller side to the matching larger side is 32.

x = 2 × 32

x = 3

Answer: 3

The figures are similar. A pair of matching sides is 4 and 6. Another side of the smaller figure is 8. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

46 = 8x

The scale factor from the smaller side to the matching larger side is 32.

x = 8 × 32

x = 12

Answer: 12

The figures are similar. A pair of matching sides is 2 and 5. Another side of the smaller figure is 4. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

25 = 4x

The scale factor from the smaller side to the matching larger side is 52.

x = 4 × 52

x = 10

Answer: 10

The figures are similar. A pair of matching sides is 4 and 10. Another side of the smaller figure is 2. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

410 = 2x

The scale factor from the smaller side to the matching larger side is 52.

x = 2 × 52

x = 5

Answer: 5

The figures are similar. A pair of matching sides is 6 and 15. Another side of the smaller figure is 4. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

615 = 4x

The scale factor from the smaller side to the matching larger side is 52.

x = 4 × 52

x = 10

Answer: 10

The figures are similar. A pair of matching sides is 12 and 6. Another side of the smaller figure is 8. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

126 = 8x

The scale factor from the smaller side to the matching larger side is 12.

x = 8 × 12

x = 4

Answer: 4

Level 3

Hard

Some of these figures are not similar. Check two ratios before you solve.

Example

The figures are similar. A pair of matching sides is 3 and 4. Another side of the smaller figure is 2. Find the matching side x.

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

34 = 2x

The scale factor from the smaller side to the matching larger side is 43.

x = 2 × 43

x = 83

Answer: 83

The figures are similar. A pair of matching sides is 3 and 5. Another side of the smaller figure is 4. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

35 = 4x

The scale factor from the smaller side to the matching larger side is 53.

x = 4 × 53

x = 203

Answer: 203

The figures are similar. A pair of matching sides is 3 and 4. Another side of the smaller figure is 5. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

34 = 5x

The scale factor from the smaller side to the matching larger side is 43.

x = 5 × 43

x = 203

Answer: 203

The figures are similar. A pair of matching sides is 2 and 5. Another side of the smaller figure is 3. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

25 = 3x

The scale factor from the smaller side to the matching larger side is 52.

x = 3 × 52

x = 152

Answer: 152

The figures are similar. A pair of matching sides is 4 and 6. Another side of the smaller figure is 3. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

46 = 3x

The scale factor from the smaller side to the matching larger side is 32.

x = 3 × 32

x = 92

Answer: 92

The figures are similar. A pair of matching sides is 5 and 2. Another side of the smaller figure is 3. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

52 = 3x

The scale factor from the smaller side to the matching larger side is 25.

x = 3 × 25

x = 65

Answer: 65

The figures are similar. A pair of matching sides is 6 and 4. Another side of the smaller figure is 5. Find the matching side x.

Show Answer

Matching sides of similar figures are proportional.

small sidematching large side = other small sidex

64 = 5x

The scale factor from the smaller side to the matching larger side is 23.

x = 5 × 23

x = 103

Answer: 103

One pair of sides is 3 to 6. Another pair is 4 to 10. Are the figures similar?

Show Answer

Figures are similar when every pair of matching sides has the same ratio.

36 and 410

12 compared with 25

The ratios are not equal, so the figures are not similar.

Answer: No

One pair of sides is 2 to 6. Another pair is 3 to 12. Are the figures similar?

Show Answer

Figures are similar when every pair of matching sides has the same ratio.

26 and 312

13 compared with 14

The ratios are not equal, so the figures are not similar.

Answer: No

One pair of sides is 5 to 10. Another pair is 4 to 6. Are the figures similar?

Show Answer

Figures are similar when every pair of matching sides has the same ratio.

510 and 46

12 compared with 23

The ratios are not equal, so the figures are not similar.

Answer: No

One pair of sides is 6 to 8. Another pair is 9 to 15. Are the figures similar?

Show Answer

Figures are similar when every pair of matching sides has the same ratio.

68 and 915

34 compared with 35

The ratios are not equal, so the figures are not similar.

Answer: No

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