Question

solve for x in the triangle. round your answer to the nearest tenth. triangle image with right angle, one leg 8, angle 54° at the vertex with leg 8, and x as the other leg

Explanation:

Step1: Identify trigonometric ratio

The triangle is right - angled, with one angle \(54^{\circ}\), adjacent side to the \(54^{\circ}\) angle is \(8\), and the side we need to find (\(x\)) is the opposite side to the \(54^{\circ}\) angle. So we use the tangent function, where \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\).
\(\tan(54^{\circ})=\frac{x}{8}\)

Step2: Solve for \(x\)

Multiply both sides of the equation by \(8\) to isolate \(x\):
\(x = 8\times\tan(54^{\circ})\)
We know that \(\tan(54^{\circ})\approx1.3764\) (using a calculator). Then \(x = 8\times1.3764 = 11.0112\)

Step3: Round to nearest tenth

Rounding \(11.0112\) to the nearest tenth gives \(11.0\) (wait, actually, \(11.0112\) rounded to the nearest tenth: the hundredth digit is \(1\), which is less than \(5\), but wait, no, \(11.0112\) is approximately \(11.0\)? Wait, no, let's recalculate \(\tan(54^{\circ})\) more accurately. \(\tan(54^{\circ})\approx1.37638192\), so \(8\times1.37638192 = 11.01105536\). Rounding to the nearest tenth: look at the hundredth place, which is \(1\), so we keep the tenths place as it is? Wait, no, \(11.01105536\) to the nearest tenth: the number is \(11.0\) when rounded? Wait, no, \(11.01\) rounded to the nearest tenth is \(11.0\)? Wait, no, \(11.01\) is closer to \(11.0\) than to \(11.1\)? Wait, no, \(11.01\) has a tenths digit of \(0\) and hundredths digit of \(1\). Since \(1<5\), we round down, so \(x\approx11.0\)? Wait, no, maybe I made a mistake in the ratio. Wait, wait, maybe the adjacent and opposite are mixed up. Wait, the right angle, the angle of \(54^{\circ}\), the side of length \(8\): is \(8\) the adjacent or opposite? Wait, the right angle is at the top, the \(54^{\circ}\) angle is at the bottom left, so the side of length \(8\) is adjacent to the \(54^{\circ}\) angle, and \(x\) is opposite. Wait, but maybe I should use cotangent? No, no, tangent is opposite over adjacent. Wait, let's check again. If the angle is \(54^{\circ}\), the side adjacent is \(8\), opposite is \(x\). So \(\tan(54)=\frac{x}{8}\), so \(x = 8\tan(54)\approx8\times1.376 = 11.0\) (rounded to the nearest tenth). Wait, but maybe I had the ratio wrong. Wait, maybe the side of length \(8\) is the opposite side? Wait, no, the right angle is at the top, so the two legs: one is \(8\) (vertical leg), one is \(x\) (horizontal leg), and the angle at the bottom is \(54^{\circ}\). So the angle at the bottom: the adjacent side to the \(54^{\circ}\) angle is the vertical leg (\(8\)), and the opposite side is the horizontal leg (\(x\)). So yes, \(\tan(54)=\frac{x}{8}\), so \(x = 8\tan(54)\approx11.0\) (rounded to the nearest tenth).

Answer:

\(11.0\) (Wait, actually, when we calculate \(8\times\tan(54^{\circ})\) with a calculator, \(\tan(54^{\circ})\approx1.376\), so \(8\times1.376 = 11.008\), which rounds to \(11.0\) when rounded to the nearest tenth. But maybe a more accurate calculation: \(\tan(54^{\circ}) = \tan(54)\approx1.37638\), so \(8\times1.37638 = 11.01104\), which rounds to \(11.0\) to the nearest tenth. )