The area of the largest triangle that can be inscribed in a semicircle whose radius is r cm, is :

Question

Accepted Answer

To determine the area of the largest triangle that can be inscribed in a semicircle with radius $ r $ cm, we need to consider the properties of such a triangle. The largest triangle that can be inscribed in a semicircle is a right-angled triangle with the diameter of the semicircle as its hypotenuse.

### Explanation for Option 2: $ r^2 \text{ cm}^2 $

1. **Properties of the Triangle:**
   - The largest triangle inscribed in a semicircle is a right triangle.
   - The hypotenuse of this triangle is the diameter of the semicircle, which is $ 2r $ cm.
   - The two other sides (legs) of the triangle are equal, as they form the maximum area when the triangle is isosceles.

2. **Area Calculation:**
   - Let the two legs of the triangle be $ a $ cm each.
   - By the Pythagorean theorem: $ a^2 + a^2 = (2r)^2 $.
   - Simplifying gives: $ 2a^2 = 4r^2 $ or $ a^2 = 2r^2 $.
   - Therefore, $ a = \sqrt{2}r $.

3. **Area of the Triangle:**
   - The area $ A $ of a right triangle is given by $ A = \frac{1}{2} \times \text{base} \times \text{height} $.
   - Here, both the base and height are $ a = \sqrt{2}r $.
   - So, $ A = \frac{1}{2} \times \sqrt{2}r \times \sqrt{2}r = \frac{1}{2} \times 2r^2 = r^2 $.

Thus, the area of the largest triangle that can be inscribed in a semicircle of radius $ r $ is $ r^2 \text{ cm}^2 $, which corresponds to Option 2.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: $ 2r \text{ cm}^2 $**
   - This option suggests a linear relationship with $ r $, which is incorrect for an area calculation. The area should be proportional to $ r^2 $, not $ r $.

2. **Option 3: $ 2r^2 \text{ cm}^2 $**
   - This option overestimates the area. The calculation shows that the maximum area is $ r^2 $, not $ 2r^2 $.

3. **Option 4: $ \frac{r^2}{2} \text{ cm}^2 $**
   - This option underestimates the area. The correct calculation yields $ r^2 $, not half of that.

In conclusion, the correct answer is Option 2: $ r^2 \text{ cm}^2 $, as it accurately represents the area of the largest triangle that can be inscribed in a semicircle with radius $ r $.