To find the area of the triangle with vertices at ((0,0)), ((1,2)), and ((3,1)), we use the shoelace formula for three points ((x_1,y_1)), ((x_2,y_2)), ((x_3,y_3)):

Step 1: Apply the shoelace formula

The formula 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| ]

Substitute the points ((0,0)), ((1,2)), ((3,1)):
[ \text{Area} = \frac{1}{2} \left| 0(2-1) + 1(1-0) + 3(0-2) \right| ]

Step 2: Calculate the expression inside the absolute value

[ 0(1) + 1(1) + 3(-2) = 0 + 1 - 6 = -5 ]

Step 3: Take absolute value and multiply by (\frac{1}{2})

[ \text{Area} = \frac{1}{2} \times |-5| = \frac{5}{2} = 2.5 ]

Answer: (\boxed{\dfrac{5}{2}}) (or (\boxed{2.5}) if decimal is allowed)

But since exact fractions are preferred in math problems, the final answer is:

(\boxed{5/2}) (or (\boxed{2.5}) depending on format, but (\boxed{5/2}) is exact).

(\boxed{5/2})

作者声明:本文包含人工智能生成内容。