Answer:
A. There is only one possible solution for this triangle.
• B=8 degrees
,• C=152 degrees
• c=13.7
Explanation:
Given the following measurements in a triangle ABC:
• a=10
,• b = 4
,• A=20°
Since the given angle A is opposite the longer side, the given measurements will produce one triangle.
(a)First, find the value of angle B using the Law of Sines.
[tex]\begin{gathered} \frac{\sin B}{b}=\frac{\sin A}{a} \\ \implies\frac{\sin B}{4}=\frac{\sin20\degree}{10} \\ Multiply\text{ both sides by 4.} \\ \sin B=4\times\frac{\sin20\degree}{10} \\ \text{Take the arcsin of both sides to solve for B.} \\ \arcsin (\sin B)=\arcsin (4\times\frac{\sin20\degree}{10}) \\ B=7.86\degree \\ Round\text{ to the nearest degree} \\ B\approx8\degree \end{gathered}[/tex](b)Next, find the value of angle C.
The sum of the measures of angles in a triangle is 180 degrees, therefore:
[tex]\begin{gathered} m\angle A+m\angle B+m\angle C=180\degree \\ Substitute\text{ the known angles} \\ 20\degree+8\degree+m\angle C=180\degree \\ 28\degree+m\angle C=180\degree \\ \text{Subtract 28 from both sides.} \\ m\angle C=180\degree-28\degree \\ m\angle C=152\degree \end{gathered}[/tex](c)Here we find the value of side length c.
Using the Law of Sines:.
[tex]\begin{gathered} \frac{c}{\sin C}=\frac{a}{\sin A} \\ \implies\frac{c}{\sin 152\degree}=\frac{10}{\sin 20\degree} \\ Multiply\text{ both sides by }\sin 152\degree\text{.} \\ c=\frac{10}{\sin20\degree}\times\sin 152\degree \\ c=13.73 \\ Round\text{ to the nearest tenth} \\ c=13.7 \end{gathered}[/tex]The values of B, C, and c are 8 degrees, 152 degrees, and 13.7 respectively.
The correct option is A.