Which of the following can be the ratio of sides of a triangle?

[BTGmoderatorDC]

Which of the following can be the ratio of sides of a triangle?

I. \(3:\sqrt{4}:5\)
II. \(\sqrt{3}:4:5\)
III. \(3:4:\sqrt{5}\)

A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III



OA D

Source: Veritas Prep

[Scott@TargetTestPrep]

Which of the following can be the ratio of sides of a triangle?

I. \(3:\sqrt{4}:5\)
II. \(\sqrt{3}:4:5\)
III. \(3:4:\sqrt{5}\)

A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III



OA D

In order to be the ratio of the sides of a triangle, the sum of the two smaller numbers should be greater than the largest number.

In I, the largest number is 5, but 3 + √4 = 3 + 2 = 5, which is not greater than 5. So I can’t be the ratio of the sides of a triangle.

In II, the largest number is also 5, and √3 + 4 ≈ 1.7 + 4 = 5.7, which is greater than 5. So II can be the ratio of the sides of a triangle.

In III, the largest number is 4, and 3 + √5 ≈ 3 + 2.2 = 5.2, which is greater than 4. So III can be the ratio of the sides of a triangle.

Answer: D