To calculate the area of a triangle given its vertices, we can use the shoelace formula (also known as the surveyor's formula), which is especially helpful for finding the area of a polygon when you have the coordinates of its vertices. The formula is:

$\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - (y_1x_2 + y_2x_3 + y_3x_1) \right|$

Where $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are the coordinates of the three vertices.

Given points:

• $(1,1)$ → $(x_1, y_1)$
• $(6,6)$ → $(x_2, y_2)$
• $(4,-2)$ → $(x_3, y_3)$

Now, substitute these values into the formula:

$\text{Area} = \frac{1}{2} \left| (1)(6) + (6)(-2) + (4)(1) - \left[(1)(6) + (6)(4) + (-2)(1)\right] \right|$

Simplifying step by step:

$\text{Area} = \frac{1}{2} \left| 6 - 12 + 4 - (6 + 24 - 2) \right|$

$\text{Area} = \frac{1}{2} \left| -2 - 28 \right|$

$\text{Area} = \frac{1}{2} \times 30 = 15$

Thus, the area of the triangle is 15 square units.

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Would you like to go over more examples of this method? Or do you have any other questions?

Here are some related questions you might find interesting:

1. How do you calculate the area of a triangle using base and height?
2. How can you determine the coordinates of the centroid of a triangle?
3. Can you calculate the area of a triangle using Heron's formula?
4. How do you find the perimeter of a triangle given its vertex coordinates?
5. What are other ways to find the area of a polygon?

Tip: The shoelace formula is very useful for polygons with any number of sides, not just triangles, as long as the vertices are listed in order around the shape.