what is the area of a triangle defined by the points A(-2,-2) B (4,-2) C(6,3)

To find the area of a triangle defined by three points in the coordinate plane, we can use the following formula:

\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]

Let the coordinates of points \( A(x_1, y_1) = (-2, -2) \), \( B(x_2, y_2) = (4, -2) \), and \( C(x_3, y_3) = (6, 3) \).

Substituting these values into the formula:

\[
\text{Area} = \frac{1}{2} \left| -2(-2 - 3) + 4(3 - (-2)) + 6((-2) - (-2)) \right|
\]
\[
= \frac{1}{2} \left| -2(-5) + 4(5) + 6(0) \right|
\]
\[
= \frac{1}{2} \left| 10 + 20 + 0 \right|
\]
\[
= \frac{1}{2} \times 30 = 15
\]

Thus, the area of the triangle is **15 square units**.

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:
1. How would the area formula change if the points were in three dimensions?
2. What happens if the points are collinear, and how would you verify that?
3. Can you derive the formula for the area of a triangle using the determinant method?
4. How does the area of a triangle relate to its perimeter in specific cases (like equilateral triangles)?
5. What is the centroid of this triangle, and how is it calculated?

**Tip:** When using this formula, ensure the order of the vertices follows a consistent direction to avoid sign errors.

What is the area of a triangle defined by the points A(-2,-2), B(4,-2), and C(6,3)?

Learn how to calculate the area of a triangle defined by points A(-2,-2), B(4,-2), and C(6,3) in the coordinate plane. This step-by-step solution covers key concepts in coordinate geometry and provides a detailed explanation using the area of a triangle formula. Suitable for students in Grades 8-10.

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