Question

post test. coordinate geometry
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 given points \((x_1,y_1)\) and \((x_1,y_2)\), the vertical distance (height) is \(y_2 - y_1\).
From the given points \((x_1,y_1)\) and \((x_3,y_1)\), the horizontal distance (base) is \(x_3 - x_1\).

Step3: Substitute into the area formula

Substituting the base \(x_3 - x_1\) and height \(y_2 - y_1\) into \(A=\frac{1}{2}\times base\times height\), we get \(A=\frac{1}{2}(y_2 - y_1)(x_3 - x_1)\).

Answer:

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