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.