Question

a. $a = \frac { 1 } { 2 } ( y _ { 3 } - y _ { 2 } ) ( x _ { 3 } - x _ { 1 } )$
b. $a = \frac { 1 } { 2 } ( y _ { 3 } - y _ { 1 } ) ( x _ { 3 } - x _ { 1 } )$
c. $a = \frac { 1 } { 2 } ( y _ { 3 } - y _ { 1 } ) ( x _ { 2 } - x _ { 1 } )$
d. $a = \frac { 1 } { 2 } ( y _ { 2 } - y _ { 1 } ) ( x _ { 3 } - x _ { 1 } )$
e. $a = \frac { 1 } { 2 } ( y _ { 2 } - y _ { 1 } ) ( x _ { 2 } - x _ { 1 } )$

Explanation:

Step1: Recall the formula for the area of a triangle

The area of a triangle is \(A=\frac{1}{2}\times base\times height\).

Step2: Identify the base and height from the coordinates

From the graph, the vertical side has endpoints \((x_1,y_1)\) and \((x_1,y_2)\), so the length of this side (height) is \(y_2 - y_1\). The horizontal distance from \(x = x_1\) to \(x=x_3\) (base) is \(x_3 - x_1\). But wait, no. Wait, actually, if we consider the vertical side \((x_1,y_1)\) and \((x_1,y_2)\) as the height \(h=y_2 - y_1\) (assuming \(y_2>y_1\)) and the base \(b=x_3 - x_1\) (horizontal distance from \(x = x_1\) to \(x=x_3\)). But no, wait, the formula \(A=\frac{1}{2}\times base\times height\). Looking at the options, we made a mistake. Wait, actually, if we consider the vertical segment from \((x_1,y_1)\) to \((x_1,y_2)\) as the height \(h = y_2 - y_1\) (assuming \(y_2>y_1\)) and the base as the horizontal distance from \(x=x_1\) to \(x = x_3\) (base \(b=x_3 - x_1\)). But wait, no. Wait, the formula for the area of a triangle with vertices \((x_1,y_1)\), \((x_1,y_2)\) and \((x_3,y_3)\): The base is the vertical segment between \((x_1,y_1)\) and \((x_1,y_2)\) with length \(|y_2 - y_1|\) and the height is the horizontal distance from \(x=x_1\) to \(x=x_3\) (length \(|x_3 - x_1|\)). But wait, no. Wait, using the formula \(A=\frac{1}{2}\times base\times height\). The base is \(y_2 - y_1\) (vertical side) and the height is \(x_3 - x_1\) (horizontal projection). So \(A=\frac{1}{2}(y_2 - y_1)(x_3 - x_1)\). But wait, no, looking at the options. Wait, no, wait, if we consider the triangle, the vertical side is from \((x_1,y_1)\) to \((x_1,y_2)\) (length \(y_2 - y_1\) if \(y_2>y_1\)) and the base (horizontal) is from \(x=x_1\) to \(x=x_3\) (length \(x_3 - x_1\)). So \(A=\frac{1}{2}(y_2 - y_1)(x_3 - x_1)\). But looking at the options, option D is \(A=\frac{1}{2}(y_2 - y_1)(x_3 - x_1)\)

Answer:

D. \(A=\frac{1}{2}(y_2 - y_1)(x_3 - x_1)\)