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


a = 50, b = 26, A = 95°

Verified step by step guidance

1. Step 1: Understand the given information. You have a triangle with sides a = 50, b = 26, and angle A = 95°.
2. Step 2: Use the Law of Sines to find angle B. The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} \).
3. Step 3: Rearrange the equation to solve for \( \sin B \): \( \sin B = \frac{b \cdot \sin A}{a} \).
4. Step 4: Calculate \( \sin B \) using the values provided: \( \sin B = \frac{26 \cdot \sin(95°)}{50} \).
5. Step 5: Determine the number of possible triangles by checking if \( \sin B \) is valid (i.e., \( 0 \leq \sin B \leq 1 \)). If valid, check if angle B and angle C (calculated as 180° - A - B) form a valid triangle.

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.

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 determining the possibility of forming a triangle with given side lengths and angles, ensuring that the sides can connect to form a closed shape.

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). In this scenario, there can be zero, one, or two possible triangles, depending on the given measurements. Understanding this case is crucial for determining how many triangles can be formed with the provided dimensions.