### Instruction: A triangle has vertices at $(-3,2),(6,-2),(3,5)$. How many square units are in the area of the triangle? Express your answer as a decimal to the nearest tenth.### Response: I want to find the area of a triangle given its three vertices.
One way to do this is to use the formula $A = \frac{1}{2}|x_1y_2 + x_2y_3 + x_3y_1 - x_1y_3 - x_2y_1 - x_3y_2|$ where $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ are the vertices.
This formula is based on the determinant of a matrix, which is related to the area of a parallelogram formed by the vectors defined by the vertices.
I can plug in the coordinates of the vertices into the formula and simplify.
$A = \frac{1}{2}|(-3)(-2) + (6)(5) + (3)(2) - (-3)(5) - (6)(2) - (3)(-2)|$
$A = \frac{1}{2}|6 + 30 + 6 + 15 - 12 + 6|$
$A = \frac{1}{2}|51|$