Given a triangle, C=100°, a=15 m, b=20 m.
a) find the distance c
b) find the area of triangle ABC

Given the triangle with \( C = 100^\circ \), \( a = 15 \, \text{m} \), and \( b = 20 \, \text{m} \), we can solve the problem using the Law of Cosines and the formula for the area of a triangle.

### Part (a): Finding the Distance \( c \)

To find side \( c \), we use the Law of Cosines:

\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]

Substituting the given values:

\[
c^2 = 15^2 + 20^2 - 2 \times 15 \times 20 \times \cos(100^\circ)
\]

Let's calculate the value of \( c \).The distance \( c \) is approximately \( 27.00 \, \text{m} \).

### Part (b): Finding the Area of Triangle ABC

To find the area of the triangle, we use the formula:

\[
\text{Area} = \frac{1}{2} ab \sin(C)
\]

Substituting the given values:

\[
\text{Area} = \frac{1}{2} \times 15 \times 20 \times \sin(100^\circ)
\]

Let's calculate the area.The area of triangle ABC is approximately \( 147.72 \, \text{m}^2 \).

### Summary:
1. Distance \( c \) is approximately \( 27.00 \, \text{m} \).
2. Area of triangle ABC is approximately \( 147.72 \, \text{m}^2 \).

Would you like more details, or do you have any questions?

### Related Questions:
1. How do you apply the Law of Cosines to different triangles?
2. What is the significance of the sine function in calculating the area of triangles?
3. How would the area change if angle \( C \) were different?
4. Can the Law of Cosines be used to find angles if all sides are known?
5. What other formulas can be used to find the area of a triangle?

**Tip:** Always ensure angles are in the correct units (degrees or radians) when using trigonometric functions.

Given a triangle, C=100°, a=15 m, b=20 m. a) find the distance c b) find the area of triangle ABC

Learn how to find the distance c and the area of triangle ABC when given the angle C=100°, and sides a=15 m, and b=20 m. This solution uses the Law of Cosines and the trigonometric area formula. Suitable for students in Grades 10-12.

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