Question

Two sides of an angle SSA of a triangle are given, determine whether the given measurements produce one triangle two triangles or no triangle at all solve each triangle that results A=10, B = 4, A=20°

Answer 1

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.

`\begin{gathered} (\sin B)/(b)=(\sin A)/(a) \\ \implies(\sin B)/(4)=(\sin20\degree)/(10) \\ Multiply\text{ both sides by 4.} \\ \sin B=4*(\sin20\degree)/(10) \\ \text{Take the arcsin of both sides to solve for B.} \\ \arcsin (\sin B)=\arcsin (4*(\sin20\degree)/(10)) \\ B=7.86\degree \\ Round\text{ to the nearest degree} \\ B\approx8\degree \end{gathered}`

(b)Next, find the value of angle C.

The sum of the measures of angles in a triangle is 180 degrees, therefore:

`\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}`

(c)Here we find the value of side length c.

Using the Law of Sines:.

`\begin{gathered} (c)/(\sin C)=(a)/(\sin A) \\ \implies(c)/(\sin 152\degree)=(10)/(\sin 20\degree) \\ Multiply\text{ both sides by }\sin 152\degree\text{.} \\ c=(10)/(\sin20\degree)*\sin 152\degree \\ c=13.73 \\ Round\text{ to the nearest tenth} \\ c=13.7 \end{gathered}`

The values of B, C, and c are 8 degrees, 152 degrees, and 13.7 respectively.

The correct option is A.