Triangles

Triangle 1: sides 3, 4, 5 and Triangle 2: sides 6, 8, 10, Triangle 1: sides 5, 12, 13 and Triangle 2: sides 10, 24, 26, Triangle 1: sides 2, 3, 4 and Triangle 2: sides 4, 6, 8

For similar triangles, the ratios of corresponding sides must be equal. Option 1: 3/6 = 4/8 = 5/10 = 1/2 ✓. Option 2: 5/10 = 12/24 = 13/26 = 1/2 ✓. Option 3: 3/6 ≠ 4/7 ≠ 5/8 ✗. Option 4: 2/4 = 3/6 = 4/8 = 1/2 ✓. Option 5: 1/2 ≠ 2/4 ≠ 3/5 ✗.

Triangle ABC ~ Triangle PQR, Corresponding angles are equal, The ratio of their areas is 4:9, The ratio of their perimeters is 2:3

Since all corresponding sides are in the ratio 2:3, by SSS similarity criterion, triangle ABC ~ triangle PQR. This means corresponding angles are equal. The triangles are not congruent as they have different sizes. The ratio of areas = (2/3)² = 4/9. The ratio of perimeters = 2/3 (same as the ratio of corresponding sides).

3 cm

By Basic Proportionality Theorem, when DE || BC, we have AD/AB = AE/AC. Given: AD = 4 cm, AB = 12 cm, AC = 9 cm. Substituting: 4/12 = AE/9. Simplifying: 1/3 = AE/9. Therefore, AE = 9 × (1/3) = 3 cm.

147 cm²

For similar triangles with sides in ratio 5:7, their areas are in the ratio (5:7)² = 25:49. If area of smaller triangle = 75 cm², then 75/Area of larger triangle = 25/49. Therefore, Area of larger triangle = (75 × 49)/25 = 3675/25 = 147 cm².