To find the area of a triangle given the coordinates of its vertices, we use the formula for the area of a triangle in coordinate geometry:

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

Here, the vertices of the triangle are:

• $(x_1, y_1) = (450, 410)$
• $(x_2, y_2) = (0, 50)$
• $(x_3, y_3) = (0, 400)$

Substitute into the formula:

$\text{Area} = \frac{1}{2} \left| 450(50 - 400) + 0(400 - 410) + 0(410 - 50) \right|$ $\text{Area} = \frac{1}{2} \left| 450(-350) + 0 + 0 \right|$ $\text{Area} = \frac{1}{2} \left| -157500 \right|$ $\text{Area} = \frac{1}{2} \cdot 157500 = 78750$

Final Answer:

The area of the triangle is 78,750 square units.

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Would you like me to break down any part of the solution further or help with similar problems? Here are 5 related questions for you:

1. How is the area of a polygon determined when more than three vertices are given?
2. What is the geometric interpretation of a negative area result in this formula?
3. How does the formula change if one vertex of the triangle lies at the origin?
4. Can you derive the formula for the area of a triangle from scratch?
5. How would this calculation change in three-dimensional space?

Tip: Always double-check coordinate inputs when using formulas in geometry to avoid small calculation errors!