Question

to find the length of ad, the necessary math needed (in order) is:
diagram of triangles: triangle abc with angle at b = 74°, angle at a = 47°, side ab = 111 m; triangle acd with angle at c = 35°, side cd = 219 m
select one:
a. primary trig ratios, cosine law
b. sine law, primary trig ratios
c. sine law, cosine law
d. cosine law, sine law

Explanation:

Step1: Analyze Triangle ABC

In triangle \(ABC\), we know angle \(B = 74^\circ\), angle \(A = 47^\circ\), and side \(AB = 111\) m. First, we can find angle \(C\) in triangle \(ABC\) using the angle - sum property of a triangle (\(\angle C=180^\circ-(74^\circ + 47^\circ)=59^\circ\)). Then, to find the length of \(AC\), we can use the Sine Law. The Sine Law states that \(\frac{AC}{\sin B}=\frac{AB}{\sin C}\). So, \(AC=\frac{AB\times\sin B}{\sin C}=\frac{111\times\sin74^\circ}{\sin59^\circ}\). This step uses the Sine Law.

Step2: Analyze Triangle ACD

In triangle \(ACD\), we know \(AC\) (found from step 1), \(CD = 219\) m, and angle \(C = 35^\circ\). To find the length of \(AD\), we can use the Cosine Law. The Cosine Law states that for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite to side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). In triangle \(ACD\), if we let \(AD = x\), \(AC = b\), \(CD=a = 219\), and the included angle between \(AC\) and \(CD\) is \(35^\circ\), then \(x^{2}=AC^{2}+219^{2}-2\times AC\times219\times\cos35^\circ\). So we first use the Sine Law in triangle \(ABC\) to find \(AC\), then use the Cosine Law in triangle \(ACD\) to find \(AD\).

Answer:

c. sine law, cosine law