The problem in the image involves a triangle $ABC$ with vertices at points $A(0, 0)$, $B(2, 3)$, and $C(3, 0)$. The task is to use the determinant method to find the area of triangle $ABC$.

Using the Determinant Formula

The area $\text{Area}$ of a triangle given its vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ can be found using the formula:

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

Substituting the Coordinates

For the triangle $ABC$ with vertices $A(0, 0)$, $B(2, 3)$, and $C(3, 0)$, the coordinates are:

• $(x_1, y_1) = (0, 0)$
• $(x_2, y_2) = (2, 3)$
• $(x_3, y_3) = (3, 0)$

Substituting these into the formula:

$\text{Area} = \frac{1}{2} \left| 0(3 - 0) + 2(0 - 0) + 3(0 - 3) \right|$

Simplifying the expression:

$\text{Area} = \frac{1}{2} \left| 0 + 0 + 3(-3) \right|$

$\text{Area} = \frac{1}{2} \left| -9 \right| = \frac{1}{2} \times 9 = 4.5$

Final Answer

The area of triangle $ABC$ is $4.5$ square units.

Would you like more details or have any questions on this method? Here are some related questions you might find interesting:

1. How does the determinant method relate to the geometric interpretation of the area of a triangle?
2. What is the importance of absolute value in the determinant formula for area?
3. Can this method be applied to find the area of a polygon with more than three sides?
4. How would the area change if one of the vertices were moved to a different coordinate?
5. What are some alternative methods to find the area of a triangle given its vertices?

Tip: The determinant method is particularly useful for finding the area when the vertices are not aligned with the coordinate axes, as it directly uses the coordinates to compute the area.