The solution you provided outlines the method to find the area of the triangle formed by the vertices A'(-1, -3), B'(-1, -3), and C'(1, -3) after determining the intersection point A' using the equations of the lines BC' and B'C. Let’s go over the steps and correct a few parts of the solution where necessary:

Step 1: Equation of Line BC'

• The line passes through points B(-2, 3) and C'(1, -3).
• The slope $m$ is calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 3}{1 - (-2)} = \frac{-6}{3} = -2$
• The point-slope form of the line is: $y - 3 = -2(x + 2)$ Simplifying the equation: $y = -2x - 1$ Correct.

Step 2: Equation of Line B'C

• The line passes through points B'(-1, -3) and C(2, 3).
• The slope $m$ is calculated as: $m = \frac{3 - (-3)}{2 - (-1)} = \frac{6}{3} = 2$
• The point-slope form of the line is: $y + 3 = 2(x + 1)$ Simplifying the equation: $y = 2x - 1$ Correct.

Step 3: Find Intersection (Point A')

• Set the two equations equal to each other: $-2x - 1 = 2x - 1$ Solve for $x$: $-2x = 2x \quad \Rightarrow \quad x = 0$
• Substitute $x = 0$ into one of the equations (e.g., $y = -2x - 1$): $y = -2(0) - 1 = -1$ So the intersection point is A'(0, -1). Correct.

Step 4: Calculate the Area of Triangle A'B'C'

• The vertices of the triangle are A'(0, -1), B'(-1, -3), and C'(1, -3).
• Using the area formula for a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$: $\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$ Substituting in the coordinates: $\text{Area} = \frac{1}{2} \left| 0((-3) - (-3)) + (-1)((-3) - (-1)) + 1((-1) - (-3)) \right|$ Simplifying each term: $\text{Area} = \frac{1}{2} \left| 0 + (-1)(-2) + 1(2) \right|$ $\text{Area} = \frac{1}{2} \left| 0 + 2 + 2 \right| = \frac{1}{2} \times 4 = 2$

Final Answer:

The area of the triangle A'B'C' is 2 square units, not 2.5 as stated in the final answer.

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Corrections:

• The final area calculation is 2 square units, not 2.5 square units.

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Would you like to go over any more details about this solution? Let me know if you have any further questions or need additional clarification!

Here are a few related questions to deepen your understanding:

1. How would you find the area of a triangle if you only had the lengths of the sides?
2. How can you verify the correctness of a line equation using a graph?
3. What are the geometric properties of the intersection of two lines in a plane?
4. How would the area change if one of the points in the triangle was moved?
5. What other methods exist for finding the area of a triangle in a coordinate plane?

Tip: When finding the area of a triangle using coordinates, always double-check the signs and terms in your area formula to avoid errors in calculation.