To calculate the area of the triangle, we can use the given data:

Known values:

• Base (b): $6.500 \, \text{cm}$
• Two equal sides (c): $5.090 \, \text{cm}$
• Angle (C): $50^\circ 19'09''$

Steps:

1. Convert the angle $50^\circ 19'09''$ into decimal degrees: $50 + \frac{19}{60} + \frac{9}{3600} = 50.3192^\circ$

2. Use the formula for the area of a triangle with two sides and the included angle: $\text{Area} = \frac{1}{2}ab \sin(C)$ Here, $a = c = 5.090 \, \text{cm}$, $b = 6.500 \, \text{cm}$, and $C = 50.3192^\circ$.

3. Calculate: $\text{Area} = \frac{1}{2} \times 5.090 \times 5.090 \times \sin(50.3192^\circ)$

Let me calculate this.import math

Given values

side_c = 5.090 # cm angle_C_deg = 50 + 19/60 + 9/3600 # Convert to decimal degrees

Convert angle to radians

angle_C_rad = math.radians(angle_C_deg)

Calculate the area of the triangle

area = 0.5 * side_c * side_c * math.sin(angle_C_rad) area_rounded = round(area, 2) # Round to two decimal places

area_rounded9.979.97successThe area of the triangle is approximately 9.97 cm².

Would you like more details on the calculation or help with another question?

Here are five related questions:

1. How do you calculate the area of a triangle using the base and height?
2. What is the difference between using degrees and radians in trigonometry?
3. Can you calculate the perimeter of the triangle based on the given dimensions?
4. How do you convert degrees, minutes, and seconds to decimal degrees?
5. How can the Law of Cosines be used to calculate the unknown sides or angles?

Tip: Always make sure your calculator is set to the correct mode (degrees or radians) for angle-based calculations.