Find the area of the triangle ABC.
Aequals40.8degrees    Bequals39.6degrees    cequals27.1 m
Question content area bottom
Part 1
What is the area of the​ triangle?

enter your response here m squared ​(Round to the nearest tenth as​ needed.)

To find the area of a triangle when two angles and one side are given, we can use the formula based on the sine rule:

\[
\text{Area} = \frac{1}{2} c^2 \times \frac{\sin(A) \sin(B)}{\sin(C)}
\]

We are given:
- \( A = 40.8^\circ \)
- \( B = 39.6^\circ \)
- \( c = 27.1 \, \text{m} \)

First, we need to find the third angle \( C \), since the sum of the angles in a triangle is \( 180^\circ \):

\[
C = 180^\circ - (A + B) = 180^\circ - (40.8^\circ + 39.6^\circ) = 99.6^\circ
\]

Now, use the formula for the area:

\[
\text{Area} = \frac{1}{2} \times c^2 \times \frac{\sin(A) \sin(B)}{\sin(C)}
\]

Substituting the values:

\[
\text{Area} = \frac{1}{2} \times (27.1)^2 \times \frac{\sin(40.8^\circ) \times \sin(39.6^\circ)}{\sin(99.6^\circ)}
\]

Now, calculate the sines of the angles and the area. I will compute this for you.The area of the triangle is approximately \( 155.1 \, \text{m}^2 \), rounded to the nearest tenth.

Do you have any further questions or would you like more details on this solution?

Here are 5 related questions to explore:
1. How can you find the third side of the triangle using the Law of Cosines?
2. What would the area be if side \( c \) was longer?
3. How does the sine rule apply when the triangle is right-angled?
4. Can you find the height of the triangle given the same information?
5. How would the area change if angle \( C \) was obtuse?

**Tip:** Always make sure to convert angles into radians when using trigonometric functions in most programming environments.

Find the area of the triangle ABC. A = 40.8 degrees, B = 39.6 degrees, c = 27.1 m. What is the area of the triangle? (Round to the nearest tenth as needed.)

Learn how to calculate the area of triangle ABC with angles A = 40.8°, B = 39.6°, and side c = 27.1 m using the sine rule. This trigonometry problem explores key concepts like the sine rule and triangle area formulas. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation