Question

which 30°-60°-90° triangle is labeled with the correct side length ratio? (three triangles with right angles and angle labels, side length labels 1, 2, √3 in different configurations)

Explanation:

Step1: Recall 30-60-90 triangle ratios

In a \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle, the side ratios are: opposite \(30^\circ\): \(1\), opposite \(60^\circ\): \(\sqrt{3}\), hypotenuse: \(2\).

Step2: Analyze each triangle

• First triangle: Right angle, \(30^\circ\) angle, side opposite \(30^\circ\) should be \(1\), but here side labeled \(1\) is adjacent to \(30^\circ\), opposite \(60^\circ\) should be \(\sqrt{3}\) (but here \(\sqrt{3}\) is adjacent to \(30^\circ\)), hypotenuse \(2\). Incorrect.
• Second triangle: Right angle, \(30^\circ\) angle: side opposite \(30^\circ\) is \(1\) (correct), opposite \(60^\circ\) is \(\sqrt{3}\) (correct), hypotenuse \(2\) (correct). Ratios match \(1 : \sqrt{3} : 2\).
• Third triangle: \(30^\circ\) angle, side opposite \(30^\circ\) should be \(1\), but here side labeled \(1\) is adjacent to \(60^\circ\), hypotenuse \(2\) (correct), but opposite \(60^\circ\) should be \(\sqrt{3}\) (here \(\sqrt{3}\) is adjacent to \(30^\circ\)). Incorrect.

Answer:

The second triangle (with angles \(30^\circ\), \(60^\circ\), right angle; sides \(1\), \(\sqrt{3}\), \(2\))