Final Answer:
The area of the triangle with side lengths 19 cm, 15 cm, and 12 cm is approximately
, calculated using Heron's formula.
Explanation:
To find the area of a triangle given its side lengths using Heron's formula, you can use the following steps:
1. Calculate the semi-perimeter (s) of the triangle using the formula:
![\[ s = (a + b + c)/(2) \]](https://img.qammunity.org/2022/formulas/mathematics/high-school/lcxjzqlp8mpym5spz3yi51y6sk84e8a6g9.png)
where (a), (b), and (c) are the side lengths of the triangle.
2. Use Heron's formula to find the area (\(A\)):
![\[ A = √(s \cdot (s - a) \cdot (s - b) \cdot (s - c)) \]](https://img.qammunity.org/2022/formulas/mathematics/high-school/r8d45g9foy852kd4lddce69zdhot5wkyqo.png)
Given the side lengths of the triangle as 19 cm, 15 cm, and 12 cm, let's calculate:
![\[ s = (19 + 15 + 12)/(2) = 23 \, \text{cm} \]](https://img.qammunity.org/2022/formulas/mathematics/high-school/ucsc7g64684sr1dzpqn6clx8pcu4mf9t06.png)
Now, plug the values into Heron's formula:
![\[ A = √(23 \cdot (23 - 19) \cdot (23 - 15) \cdot (23 - 12)) \]](https://img.qammunity.org/2022/formulas/mathematics/high-school/f24gnle84b18auf8fd3fqwjq0234d0xcas.png)
![\[ A = √(23 \cdot 4 \cdot 8 \cdot 11) \]](https://img.qammunity.org/2022/formulas/mathematics/high-school/a7v2sqzblmzacxuoiipnbry61xtlj3tj8d.png)
![\[ A = √(6472) \]](https://img.qammunity.org/2022/formulas/mathematics/high-school/ig57j1qrlnnhckztvt2ye30eqacovvkvd6.png)
![\[ A \approx 80.46 \, \text{cm}^2 \]](https://img.qammunity.org/2022/formulas/mathematics/high-school/5ocudlx21mn63gaapjvhbhyozumx8fbyk7.png)
So, the area of the triangle is approximately
.