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

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

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

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

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

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)

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

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