Question

Which of the following could be the ratio between the lengths of the two legs of a 30-60-90 triangle? Check all that apply.

Options:
Option 1: 1:1
Option 2: 1:√3
Option 3: 3:1
Option 4: 2:2√3

Answer 1

In a 30-60-90 triangle, Option 2 (1:\(\sqrt{3}\)) and Option 4 (2:2\(\sqrt{3}\)) are the correct ratios representing the lengths of the two legs, while Options 1 (1:1) and 3 (3:1) are not correct.

Step-by-step explanation:

The ratios between the lengths of the two legs of a 30-60-90 triangle are specific and are based on the properties of such a triangle. In a 30-60-90 triangle, the length of the shorter leg (opposite the 30-degree angle) is always half the hypotenuse, and the length of the longer leg (opposite the 60-degree angle) is the shorter leg times \(\sqrt{3}\). Therefore, the correct ratios for the lengths of the two legs in such a triangle would be:

• Option 2: 1:\(\sqrt{3}\) - Correct, represents the ratio between the shorter leg and the longer leg (1: \(\sqrt{3}\)).
• Option 4: 2:2\(\sqrt{3}\) - Also correct, if we consider the hypotenuse to be 2 times the shorter leg, then the longer leg would be 2 times \(\sqrt{3}\) times the shorter leg.

Options 1 (1:1) and 3 (3:1) are incorrect because they do not represent the relationship between the lengths of the legs in a 30-60-90 triangle.