Question

which of the following are not possible lengths of
the three sides of a 30-60-90 triangle?

□ 7,14,7√3

□ 3,4,5

□ 6,12,6√3

rewatch submit

Explanation:

Step1: Recall 30-60-90 triangle ratios

In a 30-60-90 triangle, the side lengths are in the ratio \(1 : \sqrt{3} : 2\), where the side opposite 30° is \(x\), opposite 60° is \(x\sqrt{3}\), and hypotenuse is \(2x\).

Step2: Check first set \(7, 14, 7\sqrt{3}\)

Let \(x = 7\). Then \(x\sqrt{3}=7\sqrt{3}\) and \(2x = 14\). This matches the ratio, so it is a valid 30-60-90 triangle.

Step3: Check second set \(3, 4, 5\)

The ratio \(3:4:5\) is a Pythagorean triple (right triangle) but not in the \(1 : \sqrt{3} : 2\) ratio. \(3^2 + 4^2 = 9 + 16 = 25 = 5^2\), but it doesn't follow \(x : x\sqrt{3} : 2x\) (e.g., if \(x = 3\), \(2x = 6
eq5\); if \(x = 4\), \(2x = 8
eq5\); if \(x = 5\), \(x\sqrt{3}=5\sqrt{3}\approx8.66
eq4\) or \(3\)). So it's not a 30-60-90 triangle.

Step4: Check third set \(6, 12, 6\sqrt{3}\)

Let \(x = 6\). Then \(x\sqrt{3}=6\sqrt{3}\) and \(2x = 12\). This matches the ratio, so it is a valid 30-60-90 triangle.

Answer:

3, 4, 5 (the option with 3, 4, 5)