Question

question 9 (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) 103.0 m
d) 123.4 m

Explanation:

Step1: Find the angle in the triangle

In the triangle with side \(225\) m, using the angle - sum property of a triangle (\(180^{\circ}\) in a triangle). Let the unknown angle be \(A\). Then \(A = 180^{\circ}-(51^{\circ}+65^{\circ})=64^{\circ}\).

Step2: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let's consider the triangle where we can relate the side of length \(225\) m and the right - triangle formed with height \(x\). First, find the length of the side adjacent to the \(27^{\circ}\) angle (let's call it \(y\)) using the Law of Sines in the non - right triangle: \(\frac{y}{\sin51^{\circ}}=\frac{225}{\sin64^{\circ}}\), so \(y=\frac{225\times\sin51^{\circ}}{\sin64^{\circ}}\). Then, in the right - triangle with angle \(27^{\circ}\), we know that \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\cos27^{\circ}=\frac{x}{y}\), so \(x = y\times\cos27^{\circ}\). Substituting \(y\) into the equation for \(x\): \(x=\frac{225\times\sin51^{\circ}\times\cos27^{\circ}}{\sin64^{\circ}}\).
Calculate \(\sin51^{\circ}\approx0.777\), \(\cos27^{\circ}\approx0.891\), \(\sin64^{\circ}\approx0.899\).
\(x=\frac{225\times0.777\times0.891}{0.899}\)
\(x=\frac{225\times0.692}{0.899}\)
\(x=\frac{155.7}{0.899}\approx173.2\) (This approach has an error. Let's use another way)

Another approach:
First, in the non - right triangle, using the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Let the side opposite to \(51^{\circ}\) be \(a\), opposite to \(64^{\circ}\) be \(225\) (since \(180-(51 + 65)=64\)), and opposite to \(65^{\circ}\) be \(b\).
\(\frac{b}{\sin65^{\circ}}=\frac{225}{\sin64^{\circ}}\), \(b=\frac{225\times\sin65^{\circ}}{\sin64^{\circ}}\approx\frac{225\times0.906}{0.899}\approx227.3\)
Then, in the right - triangle with angle \(27^{\circ}\), \(\sin27^{\circ}=\frac{x}{b}\) (where \(b\) is the hypotenuse of the right - triangle). So \(x = b\times\sin27^{\circ}\)
\(x\approx227.3\times0.454\approx103.0\)

Answer:

C. \(103.0\) m