How do I use three vertices to calculate the area of a triangle?

The three vertices are (0,0), (2,1), and (-1,6).
I've figured out that the equations of the lines that make up the triangle are y = -6x, y = (1/2)x, and y = (-5/3)x + 4.34ish.
Now I have to integrate to find out the area somehow and am confused as how to do this. Any help would be great! Thanks!

integrate?

http://mathworld.wolfram.com/TriangleArea.html

i tried but the answer i received was not 6.5, which is the correct answer.

Just integrate

a = ∫[-1,0] (13-5x)/3 - (-6x) dx
+ ∫[0,2] (13-5x)/3 - x/2 dx
= 13/6 + 13/3 = 13/2

Or

The distance from (0,0) to (1,2) is √5. Let that be the base of the triangle.

The altitude is the distance from (-1,6) to y=x/2 (or -x+2y+0=0) is

|1+12|/√5 = 13/√5

Thus, the area is 1/2 bh = (1/2)(√5)(13/√5) = 13/2

To calculate the area of a triangle using three vertices, you can use coordinate geometry and the Shoelace Formula. Here are the steps to follow:

Step 1: Plot the given vertices on a coordinate plane. In this case, the vertices are (0,0), (2,1), and (-1,6).

Step 2: Connect the vertices to form a triangle.

Step 3: Identify the coordinates of the vertices in order, either clockwise or counterclockwise. In this case, let's go clockwise: (0,0), (2,1), and (-1,6).

Step 4: Use the Shoelace Formula, given as:
Area = 0.5 * |(x1 * y2 + x2 * y3 + x3 * y1) - (x2 * y1 + x3 * y2 + x1 * y3)|

So, let's substitute the values from the given vertices:
Area = 0.5 * |(0 * 1 + 2 * 6 + (-1) * 0) - (2 * 0 + (-1) * 1 + 0 * 6)|

Simplifying further:
Area = 0.5 * |(0 + 12 + 0) - (0 + (-1) + 0)|
Area = 0.5 * |12 + 1|

Finally:
Area = 0.5 * 13
Area = 6.5

Therefore, the area of the triangle formed by the given vertices is 6.5 square units.

Note: The equations of the lines you mentioned (y = -6x, y = (1/2)x, and y = (-5/3)x + 4.34ish) are not required to calculate the area of the triangle in this case. They describe the lines that the sides of the triangle would be if they were extended.