Similar Triangles

AA, SAS, and SSS similarity theorems

Similar Triangles

Definition

Two triangles are similar if:

All corresponding angles are congruent
All corresponding sides are proportional

Symbol: $\triangle ABC \sim \triangle DEF$

Similarity Postulates

AA (Angle-Angle)

If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

Note: If two angles match, the third must also match (angle sum = 180°)

SAS (Side-Angle-Side)

If two sides of one triangle are proportional to two sides of another triangle AND the included angles are congruent, the triangles are similar.

SSS (Side-Side-Side)

If all three pairs of corresponding sides are proportional, the triangles are similar.

Scale Factor

The ratio of corresponding sides: $k = \frac{\text{side in triangle 1}}{\text{corresponding side in triangle 2}}$

Properties

If $\triangle ABC \sim \triangle DEF$ with scale factor $k$ :

Perimeters are in ratio $k:1$
Areas are in ratio $k^2:1$

Applications

Finding unknown lengths
Indirect measurement
Proving geometric relationships

📚 Practice Problems

1Problem 1easy

❓ Question:

Two triangles are similar. The sides of the first triangle are 3, 4, and 5 cm. The shortest side of the second triangle is 6 cm. Find the other two sides of the second triangle.

💡 Show Solution

Step 1: Identify corresponding sides: Shortest side of first triangle: 3 cm Shortest side of second triangle: 6 cm

Step 2: Find the scale factor: Scale factor = 6/3 = 2 The second triangle is 2 times larger

Step 3: Apply scale factor to other sides: Second side: 4 × 2 = 8 cm Third side: 5 × 2 = 10 cm

Step 4: Verify the similarity ratio: All ratios should be equal: 6/3 = 2 ✓ 8/4 = 2 ✓ 10/5 = 2 ✓

Answer: The other two sides are 8 cm and 10 cm

2Problem 2easy

❓ Question:

In $\triangle ABC$ , $\angle A = 50°$ and $\angle B = 60°$ . In $\triangle DEF$ , $\angle D = 50°$ and $\angle E = 60°$ . Are the triangles similar?

💡 Show Solution

Two angles of $\triangle ABC$ are congruent to two angles of $\triangle DEF$ :

$\angle A \cong \angle D$ (both $50°$ )
$\angle B \cong \angle E$ (both $60°$ )

By AA (Angle-Angle) similarity, the triangles are similar.

Answer: Yes, $\triangle ABC \sim \triangle DEF$ by AA

3Problem 3easy

❓ Question:

Triangle ABC is similar to triangle DEF. If AB = 12, BC = 15, and DE = 8, find EF.

💡 Show Solution

Step 1: Identify corresponding sides: Since △ABC ~ △DEF: AB corresponds to DE BC corresponds to EF

Step 2: Find the scale factor: Scale factor = DE/AB = 8/12 = 2/3

Step 3: Set up proportion for unknown side: BC/EF = AB/DE 15/EF = 12/8

Step 4: Solve for EF: 15/EF = 3/2 3 × EF = 15 × 2 3 × EF = 30 EF = 10

Step 5: Verify with scale factor: EF = BC × (2/3) = 15 × (2/3) = 10 ✓

Answer: EF = 10

4Problem 4medium

❓ Question:

If $\triangle ABC \sim \triangle XYZ$ with sides $AB = 6$ , $BC = 8$ , $AC = 10$ and $XY = 9$ , find $YZ$ and $XZ$ .

💡 Show Solution

Find the scale factor using corresponding sides: $k = \frac{XY}{AB} = \frac{9}{6} = \frac{3}{2}$

For YZ (corresponds to BC): $\frac{YZ}{BC} = \frac{3}{2}$ $\frac{YZ}{8} = \frac{3}{2}$ $YZ = 12$

For XZ (corresponds to AC): $\frac{XZ}{AC} = \frac{3}{2}$ $\frac{XZ}{10} = \frac{3}{2}$ $XZ = 15$

Answer: $YZ = 12$ , $XZ = 15$

5Problem 5medium

❓ Question:

In similar triangles, the ratio of corresponding sides is 3:5. If the perimeter of the smaller triangle is 24 cm, what is the perimeter of the larger triangle?

💡 Show Solution

Step 1: Understand the property: In similar triangles, the ratio of perimeters equals the ratio of corresponding sides

Step 2: Set up the proportion: Perimeter ratio = Side ratio P_small/P_large = 3/5

Step 3: Substitute known value: 24/P_large = 3/5

Step 4: Cross multiply: 3 × P_large = 24 × 5 3 × P_large = 120 P_large = 40

Step 5: Verify: Ratio: 24/40 = 3/5 ✓

Answer: The perimeter of the larger triangle is 40 cm

6Problem 6medium

❓ Question:

Triangle ABC has sides 6, 8, and 10. Triangle DEF has sides 9, 12, and 15. Are these triangles similar? If so, what is the scale factor?

💡 Show Solution

Step 1: Order the sides from smallest to largest: Triangle ABC: 6, 8, 10 Triangle DEF: 9, 12, 15

Step 2: Check ratios of corresponding sides: Shortest sides: 9/6 = 3/2 = 1.5 Middle sides: 12/8 = 3/2 = 1.5 Longest sides: 15/10 = 3/2 = 1.5

Step 3: Analyze the ratios: All three ratios are equal to 3/2

Step 4: Conclusion: Since all corresponding side ratios are equal, the triangles ARE similar by SSS Similarity

Step 5: Identify scale factor: Scale factor = 3/2 or 1.5 Triangle DEF is 1.5 times larger than triangle ABC

Answer: Yes, the triangles are similar with scale factor 3/2 (or 1.5)

7Problem 7hard

❓ Question:

A tree casts a shadow 24 feet long at the same time a 6-foot person casts a 4-foot shadow. How tall is the tree?

💡 Show Solution

The sun creates similar triangles (same angle).

Set up proportion: $\frac{\text{tree height}}{\text{tree shadow}} = \frac{\text{person height}}{\text{person shadow}}$

$\frac{h}{24} = \frac{6}{4}$

Solve: $4h = 144$ $h = 36$

Answer: The tree is 36 feet tall

8Problem 8hard

❓ Question:

A tree casts a shadow 24 feet long at the same time a 6-foot person casts a shadow 4 feet long. How tall is the tree?

💡 Show Solution

Step 1: Identify the similar triangles: The tree and its shadow form a right triangle The person and their shadow form a right triangle Since the sun angle is the same, these triangles are similar

Step 2: Set up the proportion: Tree height/Tree shadow = Person height/Person shadow h/24 = 6/4

Step 3: Simplify the right side: h/24 = 3/2

Step 4: Cross multiply: 2h = 24 × 3 2h = 72 h = 36

Step 5: Alternative method - find scale factor: Scale factor = 24/4 = 6 Tree height = 6 × 6 = 36 feet

Step 6: Verify: 36/24 = 3/2 ✓ 6/4 = 3/2 ✓ The ratios are equal

Answer: The tree is 36 feet tall

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