Question

1. triangle diagram: right angle, 60° angle, hypotenuse length 3, sides labeled ( a ) and ( b )

Explanation:

Assuming we need to find the lengths of \(a\) and \(b\) in this right - angled triangle (since there is a right angle and a \(60^{\circ}\) angle, the third angle is \(30^{\circ}\)):

Step 1: Analyze the angles and sides

In a right - angled triangle, if one of the non - right angles is \(60^{\circ}\) and the hypotenuse is \(c = 3\) (the side opposite the right angle), and we know the relationships in a \(30 - 60 - 90\) triangle:

• The side opposite the \(30^{\circ}\) angle (\(b\)) is half of the hypotenuse.
• The side opposite the \(60^{\circ}\) angle (\(a\)) is \(\frac{\sqrt{3}}{2}\) times the hypotenuse.

Step 2: Find the length of \(b\)

The angle adjacent to side \(b\) is \(60^{\circ}\), so the angle opposite to side \(b\) is \(30^{\circ}\). In a \(30 - 60 - 90\) triangle, the side opposite \(30^{\circ}\) (\(b\)) is given by \(b=\frac{1}{2}\times\) hypotenuse.
Since the hypotenuse \(c = 3\), then \(b=\frac{3}{2}=1.5\)

Step 3: Find the length of \(a\)

The side \(a\) is opposite the \(60^{\circ}\) angle. In a \(30 - 60 - 90\) triangle, the side opposite \(60^{\circ}\) (\(a\)) is given by \(a=\frac{\sqrt{3}}{2}\times\) hypotenuse.
Since the hypotenuse \(c = 3\), then \(a=\frac{3\sqrt{3}}{2}\approx\frac{3\times1.732}{2}=\frac{5.196}{2} = 2.598\)

If we assume the problem is to find \(b\):

Answer:

\(b = \frac{3}{2}\) (or \(1.5\))

If we assume the problem is to find \(a\):