Question

what are possible lengths, in units, of a, b, and c? express your answers in exact lengths.
a:
b:
c:

Explanation:

Step1: Identify the triangle type

This is a 30-60-90 right triangle. In such a triangle, the angles are \(90^\circ\), \(60^\circ\), and \(30^\circ\) (since the sum of angles in a triangle is \(180^\circ\), so the third angle is \(180 - 90 - 60=30^\circ\)). The side ratios are \(1 : \sqrt{3}: 2\) for the sides opposite \(30^\circ\), \(60^\circ\), and \(90^\circ\) respectively.

Let's assume the side opposite \(30^\circ\) (which is side \(a\) here, since \(a\) is adjacent to \(60^\circ\) and opposite \(30^\circ\)) has length \(x\). Then:

• Side \(a\) (opposite \(30^\circ\)): Let's take \(x = 1\) (we can choose any positive real number, here we choose 1 for simplicity). So \(a = 1\).

Step2: Find side \(b\) (opposite \(60^\circ\))

In a 30-60-90 triangle, the side opposite \(60^\circ\) (which is \(b\)) is \(\sqrt{3}\) times the side opposite \(30^\circ\) (which is \(a\)). So if \(a = 1\), then \(b=\sqrt{3}\times a=\sqrt{3}\times1 = \sqrt{3}\).

Step3: Find side \(c\) (hypotenuse, opposite \(90^\circ\))

The hypotenuse \(c\) is twice the side opposite \(30^\circ\) (which is \(a\)). So \(c = 2\times a=2\times1 = 2\).

Answer:

\(a\): \(1\) (or any positive real number \(x\), here we use \(x = 1\) as an example)
\(b\): \(\sqrt{3}\) (corresponding to \(a = 1\))
\(c\): \(2\) (corresponding to \(a = 1\))

(Note: If we choose \(a = k\) (where \(k>0\)), then \(b = k\sqrt{3}\) and \(c = 2k\). For example, if \(a = 2\), then \(b = 2\sqrt{3}\), \(c = 4\), etc.)