Question

question 10 (mandatory) (1 point)
determine the length of x, to the nearest tenth of a metre.
225 m
a) 202.2 m
b) 115.6 m
c) 123.4 m
d) 103.0 m

Explanation:

Step1: Use the Law of Sines in the base triangle

In the base triangle with sides and angles, by the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let's first find the side adjacent to the \(27^{\circ}\) angle. The angles in the base triangle: \(A = 51^{\circ}\), \(B=65^{\circ}\), \(C = 180^{\circ}-(51^{\circ}+65^{\circ})=64^{\circ}\), and the side \(c = 225\) m. Let the side adjacent to \(27^{\circ}\) (let's call it \(y\)): \(\frac{y}{\sin65^{\circ}}=\frac{225}{\sin64^{\circ}}\), so \(y=\frac{225\times\sin65^{\circ}}{\sin64^{\circ}}\approx227.5\) m.

Step2: Use trigonometry in the right - triangle

In the right - triangle (formed with height \(x\), the side \(y\) (from step 1) and the angle \(27^{\circ}\)), since \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), and here \(\theta = 27^{\circ}\), opposite side is \(x\) and hypotenuse is \(y\). So \(x = y\times\sin27^{\circ}\). Substitute \(y\approx227.5\) into the formula: \(x\approx227.5\times\sin27^{\circ}\approx227.5\times0.454\approx103.0\) m.

Answer:

d) \(103.0\) m