Question

find the length of side ( x ) to the nearest tenth.
(there is a right triangle in the picture, with one angle of ( 30^circ ), one angle of ( 60^circ ), the right - angled side adjacent to the ( 30^circ ) angle is ( x ), and the right - angled side adjacent to the ( 60^circ ) angle has a length of 3. )
answer attempt 1 out of 2
( x = ) input box
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Explanation:

Step1: Identify triangle type and trigonometric ratio

This is a right - triangle with angles \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\). We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For the \(30^{\circ}\) angle, the opposite side is \(3\) and the adjacent side is \(x\)? Wait, no. Wait, for the \(60^{\circ}\) angle, the opposite side is \(x\) and the adjacent side is \(3\). Since \(\tan(60^{\circ})=\frac{x}{3}\) (because in a right - triangle, \(\tan\) of an angle is the ratio of the length of the opposite side to the length of the adjacent side). And we know that \(\tan(60^{\circ})=\sqrt{3}\approx1.732\).

Step2: Solve for \(x\)

From \(\tan(60^{\circ})=\frac{x}{3}\), we can re - arrange the formula to solve for \(x\). Multiply both sides of the equation by \(3\): \(x = 3\times\tan(60^{\circ})\). Substitute \(\tan(60^{\circ})=\sqrt{3}\approx1.732\) into the equation: \(x=3\times1.732 = 5.196\). Rounding to the nearest tenth, we look at the hundredth place. The digit in the hundredth place is \(9\), which is greater than or equal to \(5\), so we round up the tenth place. \(5.196\approx5.2\).

Answer:

\(x = 5.2\)