Questions: Use Heron's Area Formula to find the area of the triangle. (Round your answer to two decimal places.) A=81°, b=74, c=41

Solution Steps

1. First, we need to find the length of the third side of the triangle using the Law of Cosines.
2. Once we have all three sides, we can use Heron's formula to find the area of the triangle.
3. Heron's formula requires calculating the semi-perimeter of the triangle and then using it to find the area.

Using the Law of Cosines, we find the length of side \( a \) as follows:

\[ a = \sqrt{b^2 + c^2 - 2bc \cos(A)} \]

Substituting the values:

\[ a = \sqrt{74^2 + 41^2 - 2 \cdot 74 \cdot 41 \cdot \cos(81^\circ)} \approx 78.7893 \]

The semi-perimeter \( s \) is calculated using the formula:

\[ s = \frac{a + b + c}{2} \]

Substituting the values:

\[ s = \frac{78.7893 + 74 + 41}{2} \approx 96.8947 \]

The area \( A \) of the triangle can be calculated using Heron's formula:

\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]

Substituting the values:

\[ A = \sqrt{96.8947 \cdot (96.8947 - 78.7893) \cdot (96.8947 - 74) \cdot (96.8947 - 41)} \approx 1498.3232 \]

Rounding to two decimal places, we have:

\[ A \approx 1498.32 \]

Final Answer