Question

what is the area of this triangle?

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: Identify base and height

The vertical side (height) is between \((x_1,y_1)\) and \((x_1,y_2)\), so height \( h = y_2 - y_1 \). The horizontal distance (base) from \( x_1 \) to \( x_3 \) is \( b = x_3 - x_1 \).

Step2: Apply triangle area formula

Area of a triangle is \( A=\frac{1}{2} \times \text{base} \times \text{height} \). Substituting \( b = x_3 - x_1 \) and \( h = y_2 - y_1 \), we get \( A = \frac{1}{2}(y_2 - y_1)(x_3 - x_1) \), which matches option D.

Answer:

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