Question

triangle ABC ~ triangle XYZ , where AB = 18 cm, BC = 30 cm , and CA = 42 cm . The longest side of triangle XYZ is 25.2 cm. What is the perimeter of triangle XYZ ?

Answer 1

Answer:

The perimeter of triangle XYZ is 54 cm

Step-by-step explanation:

• In similar triangles, their corresponding sides are proportional, which means the ratios of the corresponding sides are equal

• Their perimeters have the same ratio as their corresponding sides, which means P1 = r × P2, where r is the ratio of similarity

• Their areas have the square of the ratio as their corresponding sides, which means A1 = r² × A2

∵ Δ ABC is similar to Δ XYZ

∵ The longest side in Δ XYZ is 25.2 cm

→ That means its corresponding side in Δ ABC is the longest side

∵ AB = 18 cm, BC = 30 cm, CA = 42 cm

∴ The longest side in Δ ABC is CA

∴ CA is the corresponding side of the side of length 25.2 cm

∴ The ratio of similarity = $\frac{25.2}{42}$ = 0.6 ⇒ (r)

∵ The perimeter of Δ ABC = AB + BC + CA

∴ The perimeter of Δ ABC = 18 + 30 + 42

∴ The perimeter of Δ ABC = 90 cm

→ Use the 2nd fact above

∵ P of Δ XYZ = r × P of Δ ABC

∵ r = 0.6 and P of Δ ABC = 90

∴ The perimeter of Δ XYZ = 0.6 × 90

∴ The perimeter of Δ XYZ = 54 cm