Question

1. find the perimeter of \\( \triangle a b c \\). show/label all work clearly. express final answer correct to one decimal place. (12marks)

Explanation:

Step1: Find the length of \(AC\)

In the right - triangle with angle \(C = 37^{\circ}\) and height \(h = 9m\). Using the cosine function \(\cos C=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(AC=\frac{9}{\cos37^{\circ}}\). Since \(\cos37^{\circ}\approx0.7986\), then \(AC=\frac{9}{0.7986}\approx11.3m\).
Using the sine function \(\sin C=\frac{\text{opposite}}{\text{hypotenuse}}\), the base of the left - hand right - triangle \(x_1 = 9\tan37^{\circ}\). Since \(\tan37^{\circ}\approx0.7536\), then \(x_1 = 9\times0.7536 = 6.7824m\).

Step2: Find the length of \(AB\)

In the right - triangle with angle \(55^{\circ}\) and height \(h = 9m\). Using the sine function \(\sin55^{\circ}=\frac{9}{AB}\), so \(AB=\frac{9}{\sin55^{\circ}}\). Since \(\sin55^{\circ}\approx0.8192\), then \(AB=\frac{9}{0.8192}\approx11.0m\).
Using the tangent function for the right - hand right - triangle, the base \(x_2 = 9\tan55^{\circ}\). Since \(\tan55^{\circ}\approx1.4281\), then \(x_2=9\times1.4281 = 12.8529m\).

Step3: Calculate the perimeter of \(\triangle ABC\)

The perimeter \(P=AC + AB+(x_1 + x_2)\).
\(P\approx11.3+11.0+(6.7824 + 12.8529)\)
\(P\approx11.3+11.0+19.6353\)
\(P\approx41.9m\)

Answer:

The perimeter of \(\triangle ABC\) is approximately \(41.9m\)