Question

e3. a surveyor measures two sides of a triangular parcel: 180 m and 240 m with an included angle of 48°. a third point creates an angle of 67° opposite the 180 m side in an adjoining survey. check consistency by computing the third side from both configurations and comment briefly on whether the data can describe the same triangle.

Explanation:

Step1: Use the Law of Cosines for the first configuration

The Law of Cosines states that \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Let \(a = 180\), \(b=240\), and \(C = 48^{\circ}\). Then \(\cos48^{\circ}\approx0.6691\).

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Step2: Use the Law of Sines for the second configuration

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\). We know \(a = 180\), \(A = 67^{\circ}\), and \(B=48^{\circ}\). First, find the third angle \(C=180-(67 + 48)=65^{\circ}\). Then, using \(\frac{c}{\sin C}=\frac{a}{\sin A}\), \(\sin67^{\circ}\approx0.9205\), \(\sin65^{\circ}\approx0.9063\)

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Answer:

The lengths of the third side from the two configurations are approximately \(180.4\) m and \(177.9\) m. Since these values are not equal, the data cannot describe the same triangle.