Similar Triangles worksheet

1. Triangle 2 sides (a₂, b₂, c₂): 6, 8, 12

1. Every side of triangle 2 is the scale factor k times the matching side of triangle 1

   $k$$=2$

2. Multiply side a₁ by the scale factor k to get its partner a₂

   $\mathrm{a₂}$$=2\times 3$$=6$

3. Multiply side b₁ by the scale factor k to get its partner b₂

   $\mathrm{b₂}$$=2\times 4$$=8$

4. Multiply side c₁ by the scale factor k to get its partner c₂

   $\mathrm{c₂}$$=2\times 6$$=12$

5. Perimeters grow by k, but areas grow by k × k (k²), because both the base and the height are multiplied by k

   $\frac{\mathrm{Area₂}}{\mathrm{Area₁}}$$=2^2$$=4$

6. Area of triangle 2: triangle 1's area (from Heron's formula, which uses only the three sides) times k²

   $\mathrm{Area₂}$$=5.332682\times 4$$=21.330729$

7. Putting it together, the sides of triangle 2 (a₂, b₂, c₂) are

   $\text{6, 8, 12}$

2. Triangle 2 sides (a₂, b₂, c₂): 5.4, 9, 9

1. Matching sides of similar triangles share one ratio. So divide the side you know in triangle 2 by its partner b₁ to get the scale factor k

   $k$$=\frac{9}{5}$$=1.8$

2. Multiply side a₁ by the scale factor k to get its partner a₂

   $\mathrm{a₂}$$=1.8\times 3$$=5.4$

3. Multiply side c₁ by the scale factor k to get its partner c₂

   $\mathrm{c₂}$$=1.8\times 5$$=9$

4. Perimeters grow by k, but areas grow by k × k (k²), because both the base and the height are multiplied by k

   $\frac{\mathrm{Area₂}}{\mathrm{Area₁}}$$={1.8}^2$$=3.24$

5. Area of triangle 2: triangle 1's area (from Heron's formula, which uses only the three sides) times k²

   $\mathrm{Area₂}$$=7.154544\times 3.24$$=23.180723$

6. Putting it together, the sides of triangle 2 (a₂, b₂, c₂) are

   $\text{5.4, 9, 9}$

3. Result: Not similar

1. Sort each triangle's sides from shortest to longest, so the sides that would match line up

   $\text{triangle 1: 3, 4, 5; triangle 2: 4.2, 5, 6.5}$

2. Divide the shortest side of triangle 2 by the shortest side of triangle 1

   $\frac{4.2}{3}$$=1.4$

3. Divide the middle side of triangle 2 by the middle side of triangle 1

   $\frac{5}{4}$$=1.25$

4. Divide the longest side of triangle 2 by the longest side of triangle 1

   $\frac{6.5}{5}$$=1.3$

5. The ratios are not all equal, so the sides don't grow by the same amount and the triangles are not similar

   $\text{Not similar}$

4. Triangle 2 sides (a₂, b₂, c₂): 8, 10.6667, 16

1. Matching sides of similar triangles share one ratio. So divide the side you know in triangle 2 by its partner a₁ to get the scale factor k

   $k$$=\frac{8}{3}$$=2.666667$

2. Multiply side b₁ by the scale factor k to get its partner b₂

   $\mathrm{b₂}$$=2.666667\times 4$$=10.666667$

3. Multiply side c₁ by the scale factor k to get its partner c₂

   $\mathrm{c₂}$$=2.666667\times 6$$=16$

4. Perimeters grow by k, but areas grow by k × k (k²), because both the base and the height are multiplied by k

   $\frac{\mathrm{Area₂}}{\mathrm{Area₁}}$$={2.666667}^2$$=7.111111$

5. Area of triangle 2: triangle 1's area (from Heron's formula, which uses only the three sides) times k²

   $\mathrm{Area₂}$$=5.332682\times 7.111111$$=37.921296$

6. Putting it together, the sides of triangle 2 (a₂, b₂, c₂) are

   $\text{8, 10.6667, 16}$

5. Result: Not similar

1. Sort each triangle's sides from shortest to longest, so the sides that would match line up

   $\text{triangle 1: 4, 4, 5; triangle 2: 3.1, 6, 9}$

2. Divide the shortest side of triangle 2 by the shortest side of triangle 1

   $\frac{3.1}{4}$$=0.775$

3. Divide the middle side of triangle 2 by the middle side of triangle 1

   $\frac{6}{4}$$=1.5$

4. Divide the longest side of triangle 2 by the longest side of triangle 1

   $\frac{9}{5}$$=1.8$

5. The ratios are not all equal, so the sides don't grow by the same amount and the triangles are not similar

   $\text{Not similar}$

6. Triangle 2 sides (a₂, b₂, c₂): 9, 12, 15

1. Every side of triangle 2 is the scale factor k times the matching side of triangle 1

   $k$$=3$

2. Multiply side a₁ by the scale factor k to get its partner a₂

   $\mathrm{a₂}$$=3\times 3$$=9$

3. Multiply side b₁ by the scale factor k to get its partner b₂

   $\mathrm{b₂}$$=3\times 4$$=12$

4. Multiply side c₁ by the scale factor k to get its partner c₂

   $\mathrm{c₂}$$=3\times 5$$=15$

5. Perimeters grow by k, but areas grow by k × k (k²), because both the base and the height are multiplied by k

   $\frac{\mathrm{Area₂}}{\mathrm{Area₁}}$$=3^2$$=9$

6. Area of triangle 2: triangle 1's area (from Heron's formula, which uses only the three sides) times k²

   $\mathrm{Area₂}$$=6\times 9$$=54$

7. Putting it together, the sides of triangle 2 (a₂, b₂, c₂) are

   $\text{9, 12, 15}$

7. Triangle 2 sides (a₂, b₂, c₂): 7.2, 9.6, 12

1. Every side of triangle 2 is the scale factor k times the matching side of triangle 1

   $k$$=2.4$

2. Multiply side a₁ by the scale factor k to get its partner a₂

   $\mathrm{a₂}$$=2.4\times 3$$=7.2$

3. Multiply side b₁ by the scale factor k to get its partner b₂

   $\mathrm{b₂}$$=2.4\times 4$$=9.6$

4. Multiply side c₁ by the scale factor k to get its partner c₂

   $\mathrm{c₂}$$=2.4\times 5$$=12$

5. Perimeters grow by k, but areas grow by k × k (k²), because both the base and the height are multiplied by k

   $\frac{\mathrm{Area₂}}{\mathrm{Area₁}}$$={2.4}^2$$=5.76$

6. Area of triangle 2: triangle 1's area (from Heron's formula, which uses only the three sides) times k²

   $\mathrm{Area₂}$$=6\times 5.76$$=34.56$

7. Putting it together, the sides of triangle 2 (a₂, b₂, c₂) are

   $\text{7.2, 9.6, 12}$

8. Triangle 2 sides (a₂, b₂, c₂): 5.4, 7.2, 7.2

1. Every side of triangle 2 is the scale factor k times the matching side of triangle 1

   $k$$=1.8$

2. Multiply side a₁ by the scale factor k to get its partner a₂

   $\mathrm{a₂}$$=1.8\times 3$$=5.4$

3. Multiply side b₁ by the scale factor k to get its partner b₂

   $\mathrm{b₂}$$=1.8\times 4$$=7.2$

4. Multiply side c₁ by the scale factor k to get its partner c₂

   $\mathrm{c₂}$$=1.8\times 4$$=7.2$

5. Perimeters grow by k, but areas grow by k × k (k²), because both the base and the height are multiplied by k

   $\frac{\mathrm{Area₂}}{\mathrm{Area₁}}$$={1.8}^2$$=3.24$

6. Area of triangle 2: triangle 1's area (from Heron's formula, which uses only the three sides) times k²

   $\mathrm{Area₂}$$=5.562149\times 3.24$$=18.021362$

7. Putting it together, the sides of triangle 2 (a₂, b₂, c₂) are

   $\text{5.4, 7.2, 7.2}$

9. Triangle 2 sides (a₂, b₂, c₂): 8.4, 8.4, 12.6

1. Every side of triangle 2 is the scale factor k times the matching side of triangle 1

   $k$$=2.1$

2. Multiply side a₁ by the scale factor k to get its partner a₂

   $\mathrm{a₂}$$=2.1\times 4$$=8.4$

3. Multiply side b₁ by the scale factor k to get its partner b₂

   $\mathrm{b₂}$$=2.1\times 4$$=8.4$

4. Multiply side c₁ by the scale factor k to get its partner c₂

   $\mathrm{c₂}$$=2.1\times 6$$=12.6$

5. Perimeters grow by k, but areas grow by k × k (k²), because both the base and the height are multiplied by k

   $\frac{\mathrm{Area₂}}{\mathrm{Area₁}}$$={2.1}^2$$=4.41$

6. Area of triangle 2: triangle 1's area (from Heron's formula, which uses only the three sides) times k²

   $\mathrm{Area₂}$$=7.937254\times 4.41$$=35.00329$

7. Putting it together, the sides of triangle 2 (a₂, b₂, c₂) are

   $\text{8.4, 8.4, 12.6}$

10. Result: Not similar

1. Sort each triangle's sides from shortest to longest, so the sides that would match line up

   $\text{triangle 1: 2, 4, 5; triangle 2: 5, 6, 9}$

2. Divide the shortest side of triangle 2 by the shortest side of triangle 1

   $\frac{5}{2}$$=2.5$

3. Divide the middle side of triangle 2 by the middle side of triangle 1

   $\frac{6}{4}$$=1.5$

4. Divide the longest side of triangle 2 by the longest side of triangle 1

   $\frac{9}{5}$$=1.8$

5. The ratios are not all equal, so the sides don't grow by the same amount and the triangles are not similar

   $\text{Not similar}$