Question

to find the length of ad, the necessary math needed (in order) is: image of triangle with points b, a, c, d; angles at b:74°, a:47°, c:35°; sides ab:111 m, 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:

1. First, analyze triangle \(ABC\): We know angle \(B = 74^\circ\), angle \(A = 47^\circ\), so we can find angle \(C\) in \(\triangle ABC\) using the angle - sum property of a triangle (\(180^\circ-(74^\circ + 47^\circ)=59^\circ\)). Then, to find the length of \(AC\), we can use the Sine Law (\(\frac{AC}{\sin B}=\frac{AB}{\sin C}\)), where \(AB = 111\space m\), \(\angle B = 74^\circ\), and \(\angle C=59^\circ\).
2. Then, analyze triangle \(ACD\): We know \(AC\) (found from the Sine Law in \(\triangle ABC\)), \(CD = 219\space m\), and \(\angle C = 35^\circ\). To find \(AD\), we can use the Cosine Law (\(AD^{2}=AC^{2}+CD^{2}-2\cdot AC\cdot CD\cdot\cos(35^\circ)\)). So the order of math needed is Sine Law (for \(\triangle ABC\)) and then Cosine Law (for \(\triangle ACD\)).

Answer:

c. sine law, cosine law