The area of triangle ABC=50000 meter square, angel C=80°, AC=500 m, and angle A=25°
a) find the distance BC
b) find the perimeter of the triangle

To solve this problem, let's summarize the given information and apply trigonometric formulas:

- **Area of triangle ABC**: \( \text{Area} = 50000 \, \text{m}^2 \)
- **Angle C**: \( \angle C = 80^\circ \)
- **Side AC**: \( AC = 500 \, \text{m} \)
- **Angle A**: \( \angle A = 25^\circ \)

### Step-by-Step Solution

#### a) Finding the distance BC

To find the distance \( BC \), we will use the formula for the area of a triangle involving two sides and the sine of the included angle:

\[
\text{Area} = \frac{1}{2} \cdot AC \cdot BC \cdot \sin(\angle A)
\]

Substitute the given values:

\[
50000 = \frac{1}{2} \cdot 500 \cdot BC \cdot \sin(80^\circ)
\]

Now, solve for \( BC \):

\[
BC = \frac{50000 \times 2}{500 \cdot \sin(80^\circ)}
\]

Let me calculate this.The distance \( BC \) is approximately **203.09 meters**.

#### b) Finding the perimeter of the triangle

To find the perimeter, we need all three sides of the triangle. We already know \( AC = 500 \) m and \( BC \approx 203.09 \) m. Next, we will find the length of side \( AB \) using the Law of Sines:

\[
\frac{AB}{\sin(\angle C)} = \frac{AC}{\sin(\angle B)}
\]

First, let's find \( \angle B \):

\[
\angle B = 180^\circ - \angle A - \angle C = 180^\circ - 25^\circ - 80^\circ = 75^\circ
\]

Now use the Law of Sines to find \( AB \):

\[
AB = \frac{AC \cdot \sin(\angle B)}{\sin(\angle C)}
\]

Let's calculate \( AB \) and then the perimeter of the triangle.The length of side \( AB \) is approximately **490.41 meters**.

#### Perimeter of the Triangle

The perimeter of triangle ABC is:

\[
\text{Perimeter} = AB + BC + AC \approx 490.41 + 203.09 + 500 = 1193.50 \, \text{meters}
\]

### Summary

a) The distance \( BC \) is approximately **203.09 meters**.  
b) The perimeter of the triangle is approximately **1193.50 meters**.

Would you like more details on any part of the solution, or do you have any questions?

Here are 5 related questions for further exploration:

1. How would the calculation change if the area of the triangle were different?
2. What if one of the angles in the triangle was obtuse? How would that affect the solution?
3. How do you calculate the height of the triangle using the known sides and angles?
4. Can you find the circumcircle radius of this triangle?
5. How would you find the area using Heron's formula instead?

**Tip:** Remember that trigonometric identities are often helpful when working with angles in triangles.

The area of triangle ABC=50000 meter square, angle C=80°, AC=500 m, and angle A=25°. a) Find the distance BC. b) Find the perimeter of the triangle.

Solve the triangle ABC where the area is 50000 m², angle C is 80°, AC is 500 m, and angle A is 25°. Learn how to find the distance BC and the perimeter using the Law of Sines and trigonometric formulas. Suitable for Grades 10-12.

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