The area of the largest triangle that can be inscribed in a semi-circle of radius $ 6 ext{ cm}$ is:

Question

The area of the largest triangle that can be inscribed in a semi-circle of radius $ 6 	ext{ cm}$ is:

Examoogle · Accepted Answer

The correct answer is Option A: $36 	ext{ cm}^2$. Maximum Area of a Triangle Inscribed in a Semi-Circle

Step-by-Step Solution & Concepts:
Key Fact: Maximum Area of a Triangle Inscribed in a Semi-Circle

Given: Radius of the semi-circle $r = 6	ext{ cm}$

Key Formulas: $	ext{Area of largest triangle in semi-circle} = r^2$

Key Points: The diameter forms the hypotenuse/base of the right-angled triangle.; Maximum altitude in a semi-circle equals its radius $r$.

Steps: Identify the Base and Height for Maximum Area: For any triangle inscribed in a semi-circle of radius $r$, the maximum area occurs when the base is the diameter ($AB = 2r$) and the third vertex lies at the highest point of the arc, making the altitude perpendicular to the base equal to the radius ($h = r$).. Calculate the Area: $$	ext{Area} = \frac{1}{2} 	imes 	ext{Base} 	imes 	ext{Height} = \frac{1}{2} 	imes (2r) 	imes r = r^2$$ $$	ext{Area} = 6^2 = 36	ext{ cm}^2$$

Answer: Option A is correct — 36 cm²

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