Question

find the area of the given triangle. round your answer to the nearest tenth. do not round any intermediate computations. 48° 18 square units

Explanation:

Step1: Identify triangle type and formula

This is a right - triangle. The area of a right - triangle is given by \(A=\frac{1}{2}\times base\times height\). We know one leg (let's say the vertical leg) is \(18\). We need to find the other leg (the horizontal leg) using trigonometry. In a right - triangle, \(\tan\theta=\frac{opposite}{adjacent}\). Here, \(\theta = 48^{\circ}\), the opposite side to \(48^{\circ}\) is the horizontal leg (let's call it \(x\)) and the adjacent side is \(18\). So \(\tan(48^{\circ})=\frac{x}{18}\), which means \(x = 18\times\tan(48^{\circ})\).

Step2: Calculate the horizontal leg

We know that \(\tan(48^{\circ})\approx1.1106\). So \(x=18\times1.1106 = 19.9908\).

Step3: Calculate the area

Now, use the area formula for the right - triangle \(A=\frac{1}{2}\times18\times x\). Substitute \(x = 19.9908\) into the formula: \(A=\frac{1}{2}\times18\times19.9908=9\times19.9908 = 179.9172\).

Step4: Round to the nearest tenth

Rounding \(179.9172\) to the nearest tenth gives \(179.9\).

Answer:

\(179.9\) square units