Question

which triangle is a 30°-60°-90° triangle?
10
5
5√3
15
5
5√3
10
5
10√3
15
10
5√3

Explanation:

Step1: Recall 30-60-90 triangle ratios

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

Step2: Check each triangle

• First triangle: Sides \(5\), \(5\sqrt{3}\), \(10\). Let \(x = 5\). Then \(x\sqrt{3}=5\sqrt{3}\), \(2x = 10\). Matches the ratio \(1:\sqrt{3}:2\).
• Second triangle: Sides \(5\), \(5\sqrt{3}\), \(15\). \(2x\) should be \(10\) (if \(x = 5\)), but hypotenuse is \(15\). Doesn't match.
• Third triangle: Sides \(5\), \(10\sqrt{3}\), \(10\). Ratio would be \(5:10\sqrt{3}:10 = 1:2\sqrt{3}:2\), not \(1:\sqrt{3}:2\).
• Fourth triangle: Sides \(10\), \(5\sqrt{3}\), \(15\). Ratio \(10:5\sqrt{3}:15 = 2:\sqrt{3}:3\), not \(1:\sqrt{3}:2\).

Answer:

The first triangle (with sides 5, \(5\sqrt{3}\), 10)