Find the area of triangle whose two sides are 8 cm and 11 cm and perimeter is 32 cm:

Question

Accepted Answer

To find the area of the triangle with two sides measuring 8 cm and 11 cm, and a perimeter of 32 cm, we can use Heron's formula. First, we need to determine the length of the third side.

Given:
- Side a = 8 cm
- Side b = 11 cm
- Perimeter = 32 cm

Let's find the third side, c:
$$ a + b + c = 32 $$
$$ 8 + 11 + c = 32 $$
$$ c = 32 - 19 = 13 $$

Now, we have the sides of the triangle as 8 cm, 11 cm, and 13 cm.

Next, we calculate the semi-perimeter (s) of the triangle:
$$ s = \frac{a + b + c}{2} = \frac{8 + 11 + 13}{2} = \frac{32}{2} = 16 $$

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

Substitute the values:
$$ A = \sqrt{16(16-8)(16-11)(16-13)} $$
$$ A = \sqrt{16 \times 8 \times 5 \times 3} $$
$$ A = \sqrt{16 \times 120} $$
$$ A = \sqrt{1920} $$

To simplify $\sqrt{1920}$:
$$ 1920 = 16 \times 120 = 16 \times (4 \times 30) = (4 \times 4) \times (2 \times 2) \times 30 $$
$$ \sqrt{1920} = \sqrt{(4 \times 4) \times (2 \times 2) \times 30} $$
$$ = 4 \times 2 \times \sqrt{30} $$
$$ = 8\sqrt{30} $$

Thus, the area of the triangle is $8\sqrt{30} \, \text{cm}^2$, which corresponds to Option 3.

Explanation of why other options are incorrect:
1. **Option 1 (42 cm²):** This value does not match the calculated area using Heron's formula.
2. **Option 2 (30 cm²):** This value is not derived from the correct application of Heron's formula for the given side lengths.
3. **Option 4 (5√30 cm²):** This value does not match the simplified result of the area calculation.

Therefore, the correct answer is Option 3, $8\sqrt{30} \, \text{cm}^2$.