Triangle ABC, with vertices A(-7,4), B(-2,7), and C(-8,6), is drawn inside a rectangle, as shown below.

What is the area, in square units, of triangle ABC?

To find the area of triangle ABC with vertices \( A(-7, 4) \), \( B(-2, 7) \), and \( C(-8, 6) \), we can use the following formula for the area of a triangle when the coordinates of its vertices are known:

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

Substituting the coordinates \( A(-7, 4) \), \( B(-2, 7) \), and \( C(-8, 6) \):

\[
\text{Area} = \frac{1}{2} \left| -7(7 - 6) + (-2)(6 - 4) + (-8)(4 - 7) \right|
\]

Simplifying step by step:

\[
\text{Area} = \frac{1}{2} \left| -7(1) + (-2)(2) + (-8)(-3) \right|
\]

\[
\text{Area} = \frac{1}{2} \left| -7 - 4 + 24 \right|
\]

\[
\text{Area} = \frac{1}{2} \left| 13 \right|
\]

\[
\text{Area} = \frac{1}{2} \times 13 = 6.5
\]

Thus, the area of triangle ABC is \( \boxed{6.5} \) square units.

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the area change if the vertices of the triangle were different?
2. What if the triangle was not inside a rectangle—would that affect the area calculation?
3. Can you use this method to find the area of a quadrilateral if given four points?
4. How does this method compare to using the determinant method for finding the area of a triangle?
5. What happens to the area if one vertex of the triangle lies on the origin?

**Tip:** When working with triangle area formulas, double-check the order of the vertices to ensure you apply the formula correctly, as it can affect the sign of the area.

Learn how to calculate the area of triangle ABC with vertices (-7, 4), (-2, 7), and (-8, 6) using the area of a triangle formula. Detailed step-by-step solution provided.

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