The problem asks us to find the area of a triangle given the coordinates of its vertices. (1) The vertices are O(0, 0), A(1, 3), and B(2, 5). (2) The vertices are A(1, 1), B(-2, 3), and C(3, -3).

1. Problem Description

The problem asks us to find the area of a triangle given the coordinates of its vertices.

(1) The vertices are O(0, 0), A(1, 3), and B(2, 5).

(2) The vertices are A(1, 1), B(-2, 3), and C(3, -3).

2. Solution Steps

(1) To find the area of a triangle with vertices (0, 0), $(x_1, y_1)$, and $(x_2, y_2)$, we can use the formula:

Area = $\frac{1}{2} |x_1y_2 - x_2y_1|$.

In this case, $(x_1, y_1)$ = (1, 3) and $(x_2, y_2)$ = (2, 5).

Area = $\frac{1}{2} |(1)(5) - (2)(3)| = \frac{1}{2} |5 - 6| = \frac{1}{2} |-1| = \frac{1}{2}$.

(2) To find the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, we can use the formula:

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

In this case, $(x_1, y_1)$ = (1, 1), $(x_2, y_2)$ = (-2, 3), and $(x_3, y_3)$ = (3, -3).

Area = $\frac{1}{2} |1(3 - (-3)) + (-2)(-3 - 1) + 3(1 - 3)|$

= $\frac{1}{2} |1(6) + (-2)(-4) + 3(-2)|$

= $\frac{1}{2} |6 + 8 - 6| = \frac{1}{2} |8| = \frac{1}{2}(8) = 4$.

3. Final Answer

(1) The area of the triangle with vertices O(0, 0), A(1, 3), and B(2, 5) is $\frac{1}{2}$.

(2) The area of the triangle with vertices A(1, 1), B(-2, 3), and C(3, -3) is

4.