In an obtuse triangle, two sides have lengths 8 and 12, and the included angle is 150 degrees. Find the area of the triangle.

Introduction / Context:For a triangle with two known sides and an included angle, the area follows a direct formula using the sine of the included angle. This is especially convenient for non-right triangles such as obtuse triangles.

Given Data / Assumptions:

• Side a = 8, side b = 12.
• Included angle C = 150 degrees.
• Units are abstract area units; final answer should be in square units.

Concept / Approach:Area of triangle with two sides and included angle: A = (1/2) * a * b * sin(C). Use the exact sine value for 150 degrees: sin(150) = sin(30) = 1/2.

Step-by-Step Solution:

Verification / Alternative check:As a reasonableness check, the product (1/2)*a*b is 48, and multiplying by a sine between 0 and 1 gives an area below 48. Since sin(150) = 0.5, area equals 24, consistent with expectations for an obtuse case with a large included angle but moderate sides.

Why Other Options Are Wrong:

• 48 sq units: Would correspond to sin(150) = 1, not possible.
• 12 sq units or 6 sq units: Would require a smaller sine value than 0.5.

Common Pitfalls:

• Using degrees in a calculator set to radians resulting in incorrect sine values.
• Confusing 150 degrees with 30 degrees in the triangle but forgetting that the sine is the same; still the included angle is 150, so the formula remains valid with sin(150) = 0.5.

Final Answer:24 sq units