To determine the area of a triangle given three coordinates, you can use the shoelace formula (or determinant formula). The formula is:

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

Given the coordinates of the triangle:

• $A(-2, 6)$ → $(x_1, y_1)$
• $B(5, 4)$ → $(x_2, y_2)$
• $C(2, -2)$ → $(x_3, y_3)$

We can substitute the values into the formula.

Step 1: Identify the coordinates

• $x_1 = -2, y_1 = 6$
• $x_2 = 5, y_2 = 4$
• $x_3 = 2, y_3 = -2$

Step 2: Apply the shoelace formula

$\text{Area} = \frac{1}{2} \left| (-2)(4) + (5)(-2) + (2)(6) - \left[ (6)(5) + (4)(2) + (-2)(-2) \right] \right|$

Step 3: Simplify

First, calculate each part: $\text{Area} = \frac{1}{2} \left| (-8) + (-10) + 12 - \left[ 30 + 8 + 4 \right] \right|$ $\text{Area} = \frac{1}{2} \left| -6 - 42 \right|$ $\text{Area} = \frac{1}{2} \times 48$ $\text{Area} = 24$

So, the area of the triangle is 24 square units.

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Would you like more details on this method, or have any questions about the steps?

Related questions:

1. How would you calculate the area of a triangle given vertices with decimals?
2. What is the meaning of the "shoelace formula" and why it works for area calculation?
3. How do you calculate the area of a triangle using Heron's formula?
4. How can the coordinates of a triangle's centroid be determined?
5. How would you calculate the area of a triangle in 3D space?

Tip: The shoelace formula is especially useful when you have the exact coordinates of the vertices, and it’s applicable to any polygon when you can define the points in order.