Question

Find the area of the triangle whose vertices, taken in order are (-4, -2), (-3, -5) \& (3, -2).

Hint:

To find the area of a triangle given its vertices, use the formula \( \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \).

Answer 1

Step 1: Formula for the area of a triangle.
The area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Step 2: Substitute the coordinates of the vertices.
The coordinates of the vertices are:
- \((x_1, y_1) = (-4, -2)\)
- \((x_2, y_2) = (-3, -5)\)
- \((x_3, y_3) = (3, -2)\)
Substitute these values into the area formula: \[ \text{Area} = \frac{1}{2} \left| (-4)((-5) - (-2)) + (-3)((-2) - (-2)) + (3)((-2) - (-5)) \right| \] \[ \text{Area} = \frac{1}{2} \left| (-4)(-3) + (-3)(0) + (3)(3) \right| \] \[ \text{Area} = \frac{1}{2} \left| 12 + 0 + 9 \right| = \frac{1}{2} \times 21 = 10.5 \] Thus, the area of the triangle is 10.5 square units.