Find the area of an obtuse-angled triangle where two sides are 8 units and 12 units, and the included angle between them is 150 degrees. What is the area (in square units)?

Introduction / Context:
This question tests the triangle area formula using two sides and the included angle. When you know two sides a and b and the angle C between them, area can be found by: Area = (1/2)*a*b*sin(C). This formula works for acute, right, and obtuse triangles. Even though the angle is 150 degrees (obtuse), the sine value is still positive, so the area remains positive. A helpful fact is sin(150 degrees) = sin(30 degrees) = 1/2. Using this identity makes the calculation quick and exact.

Given Data / Assumptions:

• Side a = 8 units
• Side b = 12 units
• Included angle C = 150 degrees
• Area formula: A = (1/2)*a*b*sin(C)

Concept / Approach:
Use A = (1/2)*a*b*sin(C). Replace sin(150) with 1/2, multiply the numbers, and express the final area in square units.

Step-by-Step Solution:
A = (1/2) * 8 * 12 * sin(150) sin(150) = sin(30) = 1/2 A = (1/2) * 96 * (1/2) A = 48 * (1/2) = 24 sq units

Verification / Alternative check:
Since 150 degrees is obtuse, the height relative to side 12 is 8*sin(150)=8*(1/2)=4. Area = (1/2)*base*height = (1/2)*12*4 = 24 sq units, matching the result.

Why Other Options Are Wrong:
48 occurs if sin(150) is incorrectly taken as 1. 12 and 6 come from missing one of the (1/2) factors. 36 comes from using a wrong sine value or wrong base-height pairing.

Common Pitfalls:
Using cos instead of sin, treating sin(150) as negative, or forgetting the initial (1/2) factor in the formula.

Final Answer:
The area of the triangle is 24 sq units.