Question

#11 find the perimeter of the triangle to nearest tenth. show all equations used to find all sides. (a right triangle with one leg 4 ft and an angle of 58° is shown)

Explanation:

Step1: Identify the triangle type

This is a right - triangle with one leg \(a = 4\) ft, and an angle \(\theta=58^{\circ}\). Let the other leg be \(b\) and the hypotenuse be \(c\).

Step2: Find the other leg \(b\)

We know that in a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\tan(58^{\circ})=\frac{4}{b}\) (since the side of length 4 is opposite the \(58^{\circ}\) angle and \(b\) is adjacent to the \(58^{\circ}\) angle).
So, \(b=\frac{4}{\tan(58^{\circ})}\). We know that \(\tan(58^{\circ})\approx1.6003\), then \(b = \frac{4}{1.6003}\approx2.5\) ft.

Step3: Find the hypotenuse \(c\)

We use the cosine function. \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos(58^{\circ})=\frac{b}{c}\), and \(c=\frac{b}{\cos(58^{\circ})}\). Also, we can use \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\sin(58^{\circ})=\frac{4}{c}\), so \(c = \frac{4}{\sin(58^{\circ})}\). Since \(\sin(58^{\circ})\approx0.8480\), then \(c=\frac{4}{0.8480}\approx4.7\) ft.

Step4: Calculate the perimeter \(P\)

The perimeter of a triangle is \(P=a + b + c\). Substituting \(a = 4\), \(b\approx2.5\), \(c\approx4.7\) into the formula, we get \(P=4 + 2.5+4.7 = 11.2\) ft.

Answer:

The perimeter of the triangle to the nearest tenth is \(\boldsymbol{11.2}\) feet.