This problem indicates a method for calculating the area of a triangle when the coordina...

This problem indicates a method for calculating the area of a triangle when the coordinates of the three vertices are given.

(a) Calculate the area of in the figure.

Hint: First calculate the area of the rectangle enclosing , and then subtract the areas of the three right triangles.

(b) Calculate the area of the triangle with vertices (1, 3), (4, 1), and (10, 4). Hint:Work with an enclosing rectangle and three right triangles, as in part (a).

(c) Using the same technique that you used in parts (a) and (b), show that the area of the triangle in the following figure is given by

Remark: If we use absolute value signs instead of the parentheses, then the formula will hold regardless of the relative positions or quadrants of the three vertices. Thus the area of a triangle with vertices

(x₁,y₁),(x₂,y₂),(x₃,y₃) is given by