Question

Area of the triangle with vertices \( (-2, 2), (1, 5) \) and \( (6, -1) \) is:

• A. \( 15 \)
• B. \( \frac{3}{5} \)
• C. \( \frac{29}{2} \)
• D. \( \frac{33}{2} \)
• E. \( \frac{35}{2} \)

Hint:

Always ensure you use the absolute value at the end. Area is a scalar quantity and cannot be negative.

Answer 1

The Correct Option is D

Concept: The area of a triangle with vertices \( (x_1, y_1), (x_2, y_2), \) and \( (x_3, y_3) \) can be calculated using the coordinate geometry formula: \[ \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \]

Step 1: Inserting the coordinates.
Using points \( (-2, 2), (1, 5), \) and \( (6, -1) \): \[ \text{Area} = \frac{1}{2} |(-2)(5 - (-1)) + (1)(-1 - 2) + (6)(2 - 5)| \] \[ = \frac{1}{2} |(-2)(6) + (1)(-3) + (6)(-3)| \]

Step 2: Simplifying the arithmetic.
\[ \text{Area} = \frac{1}{2} |-12 - 3 - 18| = \frac{1}{2} |-33| \] \[ \text{Area} = \frac{33}{2} \]