Question

1. triangle with right angle, 30° angle, and side labeled ( 9x - 3 ) o. solve for x.

Explanation:

Step1: Identify triangle type

This is a right - triangle with one angle \(30^{\circ}\). In a right - triangle, the sum of angles is \(180^{\circ}\), and the right - angle is \(90^{\circ}\), the given acute angle is \(30^{\circ}\), so the other acute angle is \(180 - 90 - 30=60^{\circ}\)? Wait, no, wait. Wait, in a 30 - 60 - 90 triangle, the side opposite \(30^{\circ}\) is half the hypotenuse, but wait, maybe the side \(9x - 3\) is opposite the \(30^{\circ}\) angle? Wait, no, wait, the right angle is one angle, so the two acute angles are \(30^{\circ}\) and \(60^{\circ}\)? Wait, no, wait, maybe the side \(9x - 3\) is the side opposite the \(30^{\circ}\) angle, and in a right - triangle, if we consider the side opposite \(30^{\circ}\), but wait, maybe the triangle is a 30 - 60 - 90 triangle, and the side \(9x - 3\) is the side opposite the \(30^{\circ}\) angle, but actually, in a right - triangle, the sum of angles: \(90^{\circ}+30^{\circ}+\text{third angle}=180^{\circ}\), so third angle is \(60^{\circ}\). But maybe the side \(9x - 3\) is related to the 30 - 60 - 90 triangle ratios. Wait, no, maybe the side \(9x - 3\) is the length of the side opposite the \(30^{\circ}\) angle, and in a right - triangle, if we assume that the hypotenuse is related, but wait, maybe the triangle is a right - triangle with angles \(30^{\circ}\), \(90^{\circ}\), so the third angle is \(60^{\circ}\), but maybe the side \(9x - 3\) is the side adjacent to the \(30^{\circ}\) angle? Wait, no, maybe the problem is that in a right - triangle, the angle sum is \(180^{\circ}\), so \(90 + 30+(9x - 3)\)? No, wait, no, the angles are \(30^{\circ}\), \(90^{\circ}\), and the third angle. Wait, no, the expression \(9x - 3\) is probably an angle? Wait, that makes more sense! Oh, right, maybe \(9x - 3\) is the measure of the third angle. In a triangle, the sum of interior angles is \(180^{\circ}\). So for a right - triangle (one angle \(90^{\circ}\)), one angle \(30^{\circ}\), and the third angle \(9x - 3\). So:

\(90+30+(9x - 3)=180\)

Step2: Solve the equation

First, simplify the left - hand side:

\(90 + 30-3+9x=180\)

\(117 + 9x=180\)

Subtract 117 from both sides:

\(9x=180 - 117\)

\(9x = 63\)

Divide both sides by 9:

\(x=\frac{63}{9}=7\)

Answer:

\(x = 7\)