The three sides of a triangle are 10 meter, 12 meter and 14 meter. The area of the incircle of that triangle is :

Question

Accepted Answer

To determine the area of the incircle of a triangle with sides 10 meters, 12 meters, and 14 meters, we first need to calculate the semi-perimeter of the triangle and then use the formula for the area of the incircle.

1. **Calculate the semi-perimeter (s) of the triangle:**

The semi-perimeter $ s $ is given by:
   $$
   s = \frac{a + b + c}{2}
   $$
   where $ a = 10 $, $ b = 12 $, and $ c = 14 $.

$$
   s = \frac{10 + 12 + 14}{2} = \frac{36}{2} = 18 \text{ meters}
   $$

2. **Calculate the area (A) of the triangle using Heron's formula:**

Heron's formula for the area of a triangle is:
   $$
   A = \sqrt{s(s-a)(s-b)(s-c)}
   $$

Substituting the values:
   $$
   A = \sqrt{18(18-10)(18-12)(18-14)}
   $$
   $$
   A = \sqrt{18 \times 8 \times 6 \times 4}
   $$
   $$
   A = \sqrt{18 \times 192}
   $$
   $$
   A = \sqrt{3456}
   $$
   $$
   A = 58.7877 \text{ square meters (approximately)}
   $$

3. **Calculate the radius (r) of the incircle:**

The radius $ r $ of the incircle is given by:
   $$
   r = \frac{A}{s}
   $$

$$
   r = \frac{58.7877}{18} \approx 3.266 \text{ meters}
   $$

4. **Calculate the area of the incircle:**

The area of the incircle is given by:
   $$
   \text{Area of incircle} = \pi r^2
   $$

$$
   \text{Area of incircle} = \pi \times (3.266)^2
   $$
   $$
   \text{Area of incircle} \approx 33.523 \text{ square meters}
   $$

This value is approximately equal to 33$\frac{11}{21}$ square meters, which corresponds to Option 1.

**Explanation of why other options are incorrect:**

- **Option 2 (22$\frac{11}{21}$ square meters):** This value is significantly lower than the calculated area of the incircle, which is approximately 33.523 square meters. Therefore, it cannot be correct.

- **Option 3 (11$\frac{11}{21}$ square meters):** This value is even lower than Option 2 and is far from the calculated area. Hence, it is incorrect.

- **Option 4 (9$\frac{11}{21}$ square meters):** This value is the smallest among the options and is much less than the calculated area of the incircle. Thus, it is incorrect.

In conclusion, the correct answer is Option 1, as it closely matches the calculated area of the incircle based on the given side lengths of the triangle.