https://file.aiquickdraw.com/mathbot/file/1724724353974psai1hd3.jpg

To solve the problem given in the image, we need to find the area of triangle \(\Delta ABC\), where:

- \(A = (2, 1)\),
- \(B = (5, 3)\),
- \(C\) is the reflection of \(B\) across the x-axis.

### Step 1: Determine the coordinates of point \(C\)
The reflection of a point \((x, y)\) across the x-axis results in the point \((x, -y)\). Therefore, the coordinates of point \(C\) will be:
\[
C = (5, -3)
\]

### Step 2: Use the formula for the area of a triangle
The area \(A\) of a triangle formed by three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Substituting the coordinates of points \(A(2, 1)\), \(B(5, 3)\), and \(C(5, -3)\) into the formula:
\[
\text{Area} = \frac{1}{2} \left| 2(3 - (-3)) + 5((-3) - 1) + 5(1 - 3) \right|
\]

### Step 3: Simplify the expression
\[
\text{Area} = \frac{1}{2} \left| 2(6) + 5(-4) + 5(-2) \right|
\]
\[
\text{Area} = \frac{1}{2} \left| 12 - 20 - 10 \right|
\]
\[
\text{Area} = \frac{1}{2} \left| -18 \right| = \frac{1}{2} \times 18 = 9
\]

### Final Answer
The area of triangle \(\Delta ABC\) is \(9\) square units.

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Would you like to go over the details of any step or have any further questions?

Here are 5 related questions for further exploration:

1. How would the area change if \(C\) was the reflection of \(B\) across the y-axis instead?
2. What is the length of the base \(BC\) in the triangle \(\Delta ABC\)?
3. If point \(A\) was moved to \((2, -1)\), how would that affect the area of the triangle?
4. Can you find the centroid of triangle \(\Delta ABC\)?
5. How would you calculate the area if the triangle was plotted in 3D space with the z-coordinates of each point given?

**Tip:** The reflection of a point across the x-axis changes only the sign of the y-coordinate, while the x-coordinate remains unchanged.

Learn how to calculate the area of triangle ABC using coordinates (2, 1), (5, 3), and the reflection point (5, -3). This problem involves coordinate geometry and the formula for the area of a triangle. Suitable for Grades 9-10.

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