Question

How do I calculate the area of a triangle when given the coordinates of its vertices?

Answer 1

To calculate the area of a triangle given the coordinates of its vertices, you can use the following formula based on the determinant of a matrix derived from the coordinates. If the vertices of the triangle are given by the coordinates \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\), the area \(A\) can be calculated using the formula:

\[
A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]

Here's a step-by-step breakdown of the calculation:

1. **Identify the coordinates** of the vertices:
- Vertex 1: \((x_1, y_1)\)
- Vertex 2: \((x_2, y_2)\)
- Vertex 3: \((x_3, y_3)\)

2. **Substitute the coordinates** into the formula.

3. **Calculate the expression** inside the absolute value.

4. **Take the absolute value** of the result.

5. **Multiply by \( \frac{1}{2} \)** to obtain the area.

### Example:

Let's say you have a triangle with vertices at \(A(1, 2)\), \(B(4, 5)\), and \(C(7, 2)\).

1. Substitute the coordinates into the formula:
\[
A = \frac{1}{2} \left| 1(5 - 2) + 4(2 - 2) + 7(2 - 5) \right|
\]

2. Simplify the expression:
\[
A = \frac{1}{2} \left| 1 \cdot 3 + 4 \cdot 0 + 7 \cdot (-3) \right|
\]
\[
A = \frac{1}{2} \left| 3 + 0 - 21 \right|
\]
\[
A = \frac{1}{2} \left| -18 \right|
\]
\[
A = \frac{1}{2} \cdot 18 = 9
\]

So, the area of the triangle is \(9\) square units.