Determine the number of triangles ABC possible with the given parts.


a = 31, b = 26, B = 48°

Verified step by step guidance

1. Identify the given parts: side \(a = 31\), side \(b = 26\), and angle \(B = 48^\circ\).
2. Use the Law of Sines to find angle \(A\): \(\frac{a}{\sin A} = \frac{b}{\sin B}\).
3. Rearrange the equation to solve for \(\sin A\): \(\sin A = \frac{a \cdot \sin B}{b}\).
4. Calculate \(\sin A\) and determine if \(\sin A\) is less than or equal to 1 to check for possible triangles.
5. If \(\sin A\) is valid, use the inverse sine function to find angle \(A\) and determine the number of possible triangles (0, 1, or 2) based on the ambiguous case of the Law of Sines.

Key Concepts

Law of Sines

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles. It states that the ratio of a side length to the sine of its opposite angle is constant for all three sides of the triangle. This law is particularly useful for solving triangles when given two angles and one side or two sides and a non-included angle.

Ambiguous Case of the Law of Sines

The ambiguous case occurs when using the Law of Sines with two sides and a non-included angle (SSA condition). Depending on the given values, there can be zero, one, or two possible triangles. Understanding this case is crucial for determining how many triangles can be formed with the provided measurements.

Triangle Inequality Theorem

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem is essential for verifying the feasibility of a triangle's existence based on its side lengths. It helps in determining whether the given sides can form a valid triangle.