Question

1 multiple choice 1 point
in \\( \triangle l m n, m \angle l = 108 ^ { \circ }, m \angle m = 25 ^ { \circ } \\) and \\( \ell = 12 \\). what is \\( m \\)?
27
about 0.19
about 5.3
about 20.8
2 fill in the blank 1 point
find the angle measure(s). round to the nearest tenth if needed.
in \\( \triangle w x y, m \angle x = 104 ^ { \circ }, x = 7 \\), and \\( y = 2 \\). find \\( m \angle y \\).
the \\( m \angle y = \\) type your answer...
3 fill in the blank 1 point
what is the length of \\( y \\)?
round to the nearest tenth. \\( y \approx \\) type your answer...

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that in a triangle \( \triangle ABC\) (or \( \triangle LMN\) here), \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). For \(\triangle LMN\), we have \(\frac{\ell}{\sin L}=\frac{m}{\sin M}\).

Step2: Substitute the given values

We know that \(\ell = 12\), \(m\angle L=108^{\circ}\), \(m\angle M = 25^{\circ}\). So, \(m=\frac{\ell\times\sin M}{\sin L}\).

Step3: Calculate the sine values

\(\sin(108^{\circ})\approx0.9511\), \(\sin(25^{\circ})\approx0.4226\).

Step4: Compute \(m\)

\(m=\frac{12\times0.4226}{0.9511}=\frac{5.0712}{0.9511}\approx5.3\)

Answer:

about \(5.3\)