Question

question
find the length of side ( x ) to the nearest tenth.

answer attempt 1 out of 2
( x = )

Explanation:

Step1: Identify the triangle type and trigonometric ratio

This is a right - triangle. We know one angle is \(30^{\circ}\), one angle is \(60^{\circ}\) and one side (opposite to \(60^{\circ}\) or adjacent? Wait, the side with length 9: let's see the angles. The right - angle, \(60^{\circ}\) and \(30^{\circ}\). Let's use the sine or cosine or tangent. Wait, in a right - triangle, for the \(30^{\circ}\) angle, the side opposite is 9? Wait no, wait the side of length 9: let's check the angles. The angle of \(30^{\circ}\), the side adjacent to \(30^{\circ}\) or opposite? Wait, let's label the triangle. Let the right - angle be \(C\), \(60^{\circ}\) be \(A\), \(30^{\circ}\) be \(B\). Then side opposite \(A(60^{\circ})\) is \(BC = 9\)? Wait no, wait the side with length 9: let's use the Law of Sines. In any triangle, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).

Wait, the hypotenuse is \(x\), the side opposite \(30^{\circ}\) is 9? Wait no, wait the angle of \(30^{\circ}\): the side opposite \(30^{\circ}\) in a 30 - 60 - 90 triangle is the shortest side. Wait, maybe we can use the Law of Sines. The triangle has angles \(90^{\circ}\), \(60^{\circ}\), \(30^{\circ}\). So \(\frac{x}{\sin90^{\circ}}=\frac{9}{\sin60^{\circ}}\). Because \(\sin90^{\circ} = 1\), and \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\approx0.866\).

Step2: Solve for \(x\)

From \(\frac{x}{\sin90^{\circ}}=\frac{9}{\sin60^{\circ}}\), we can re - arrange to get \(x=\frac{9}{\sin60^{\circ}}\).

Substitute \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\approx0.866\) into the formula:

\(x=\frac{9}{\frac{\sqrt{3}}{2}}=\frac{18}{\sqrt{3}} = 6\sqrt{3}\approx6\times1.732 = 10.4\) (rounded to the nearest tenth).

Wait, alternatively, if we consider the angle of \(30^{\circ}\), the side adjacent to \(30^{\circ}\) is 9? Wait no, maybe I made a mistake. Wait, let's check the angles again. The triangle has a right - angle, \(60^{\circ}\) and \(30^{\circ}\). Let's denote the sides: let the side opposite \(30^{\circ}\) be \(a\), opposite \(60^{\circ}\) be \(b\), hypotenuse be \(c\). Then \(a=\frac{c}{2}\), \(b = \frac{c\sqrt{3}}{2}\). Wait, if the side of length 9 is opposite \(60^{\circ}\) (i.e., \(b = 9\)), then \(b=\frac{c\sqrt{3}}{2}\), so \(c=\frac{2b}{\sqrt{3}}=\frac{2\times9}{\sqrt{3}}=\frac{18}{\sqrt{3}} = 6\sqrt{3}\approx10.4\), which matches the previous result.

Answer:

\(10.4\)