The image shows a triangle with labeled vertices and asks for the area of the triangle. The problem seems to involve calculating the area based on given coordinates, possibly applying a formula.

To find the area of a triangle with vertices at $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, the formula used is:

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

Let me extract the coordinates from the graph and calculate the area of the triangle. I'll review the image closely and provide a detailed solution.I've opened the image and inspected it closely. It appears that the triangle's vertices are labeled with their coordinates. Now, I'll extract those coordinates from the image and apply the area formula to solve the problem.

Let me continue by analyzing the exact coordinates of the triangle and calculating the area.The triangle in the image has vertices labeled as:

• $A(1, 3)$
• $B(4, 1)$
• $C(2, 4)$

Using the formula for the area of a triangle with coordinates $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$:

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

Substituting the values for points $A$, $B$, and $C$:

$\text{Area} = \frac{1}{2} \left| 1(1 - 4) + 4(4 - 3) + 2(3 - 1) \right|$

Now, I will calculate this expression.The area of the triangle with vertices at $A(1, 3)$, $B(4, 1)$, and $C(2, 4)$ is $2.5$ square units.

Would you like further details on the calculation or have any other questions?

Here are five related questions to expand on this concept:

1. How does the area formula for triangles change in 3D space?
2. Can you use the distance formula to verify if the triangle's sides form a right triangle?
3. What happens if one of the vertices of the triangle coincides with the origin?
4. How can you use matrix determinants to find the area of a triangle?
5. What is the significance of the absolute value in the area formula?

Tip: When working with triangles on a coordinate plane, using the determinant of a matrix can offer an efficient method for calculating the area!