Given a triangle where $a$ = $7$, $b$ = $10$, and angle $A$ = $30°$, use the Law of Sines to determine the type of triangle that can be formed.

• A. There is only one possible triangle (unique solution)
• B. There are two possible triangles (an ambiguous case, SSA)
• C. No triangle can be formed with the given information
• D.

Answer: (B)

Verified step by step guidance

1. Identify the given elements: side \( a = 7 \), side \( b = 10 \), and angle \( A = 30^\circ \). Note that angle \( A \) is opposite side \( a \).
2. Recall the Law of Sines formula: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} \] This relates the sides and their opposite angles in any triangle.
3. Use the Law of Sines to find \( \sin(B) \): \[ \sin(B) = \frac{b \sin(A)}{a} = \frac{10 \times \sin(30^\circ)}{7} \] Calculate the value inside the sine function but do not find the angle yet.
4. Analyze the value of \( \sin(B) \): - If \( \sin(B) > 1 \), no triangle is possible. - If \( \sin(B) = 1 \), there is exactly one right triangle. - If \( 0 < \sin(B) < 1 \), there can be one or two possible triangles (ambiguous case). Determine which case applies here based on the value from step 3.
5. If two triangles are possible, find the two possible angles for \( B \) using: \[ B_1 = \arcsin(\sin(B)) \quad \text{and} \quad B_2 = 180^\circ - B_1 \] Then check if the sum of angles \( A + B_2 < 180^\circ \) to confirm the second triangle's validity.