A triangle is inscribed in a semicircle of radius 6 m. What is the area of the largest possible triangle that can be inscribed in this semicircle?

Introduction / Context:
This question explores the classic result about the largest triangle that can be inscribed in a semicircle. It builds on two key ideas: first, that any angle subtended by the diameter in a semicircle is a right angle, and second, that the area of a triangle depends on its base and corresponding height. The problem asks for the maximum area, which occurs for a specific configuration of the triangle inside the semicircle.

Given Data / Assumptions:

• Radius of the semicircle r = 6 m.
• Diameter of the semicircle is 2r = 12 m.
• The triangle is completely contained in the semicircle and one side lies along the diameter for the maximal area case.
• The largest area occurs when the triangle is right angled with the hypotenuse along the diameter.

Concept / Approach:
A key theorem states that any angle in a semicircle is a right angle. When a triangle is inscribed in a semicircle with the diameter as one side, the vertex opposite the diameter lies on the semicircle, and the triangle is right angled at that vertex. Among all such triangles, the one with the largest area has its right angle at the point of the semicircle such that the triangle height equals the radius. Thus, the base is the diameter and the maximum height is the radius, giving a simple formula for the maximum area.

Step-by-Step Solution:
Radius r = 6 m, so diameter = 2r = 12 m. For the largest triangle, take the diameter as base and the radius as height. Base b = 12 m, height h = 6 m. Area of a triangle = (1/2) * base * height. Area = (1/2) * 12 * 6 = 6 * 6 = 36 sq m.

Verification / Alternative check:
If you attempt any other triangle in the semicircle whose base is shorter than the diameter or whose height is less than the radius, the product base * height becomes smaller, and hence the area decreases. Therefore, the combination base = 12 m and height = 6 m indeed yields the maximum possible area, confirming 36 square metres as the correct answer.

Why Other Options Are Wrong:
72 sq m: This is double the correct value and would require base or height values not compatible with a semicircle of radius 6 m. 18 sq m and 12 sq m: These correspond to smaller base or height values and do not represent the maximum area triangle. 24 sq m: Also smaller than 36 sq m and results from taking a reduced effective height or base.

Common Pitfalls:
Forgetting that the largest triangle is right angled with the diameter as hypotenuse. Using the radius incorrectly as the base rather than as the height. Misapplying area formulas for sectors or segments instead of the triangle formula.

Final Answer:
The maximum possible area of the triangle is 36 sq m