Determine the number of triangles ABC possible with the given parts.
-b = 42, c = 52, B = 106°
A)0
B)1
C)3
D)2

Question

Quizplus · Accepted Answer

Given b = 42, c = 52, and B = 106°, we can use the Law of Sines to determine if a triangle is possible. The Law of Sines states that (sin A / a) = (sin B / b) = (sin C / c). However, in this case, we are given two sides and an angle not between them (SSA condition), which can lead to different scenarios. To check for the existence of a triangle, we can calculate the length of side a using the Law of Sines. However, given that angle B is greater than 90°, the sum of angles B and C must be less than 180°, which leaves a very small range for angle C (since all internal angles of a triangle must sum up to 180°). This specific setup, with the given lengths and the large angle, might not allow for the formation of a valid triangle because the side opposite the given angle might not be long enough to meet the other side (forming a triangle) due to the constraints of the triangle inequality theorem.Upon further analysis, considering the specific values given and the large angle, it becomes apparent that there is no configuration that satisfies the triangle inequality theorem and allows for a triangle to be formed. The large angle B significantly restricts the possible configurations, leading to the conclusion that no triangle can be formed with the given measurements.

Determine the Number of Triangles ABC Possible with the Given