91Ó°ÊÓ

The determinant formula is a very useful way to calculate the area of a triangle when you know its vertices. This method avoids needing to draw the triangle or use trigonometry. The formula uses the coordinates of the vertices. If you have a triangle with vertices at \((x_1, y_1), (x_2, y_2), \text{ and } (x_3, y_3)\), you can calculate its area with the formula: \[ \text{Area} = \frac{1}{2} \big| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \big| \] This notation comes from linear algebra where determinants define the area of parallelograms and half that for triangles.

Let's break this down further. You are taking each x-coordinate, multiplying it by the difference of the other two y-coordinates, and then adding them all. The absolute value ensures we always get a non-negative area. Finally, divide by 2 since a triangle is half of a parallelogram.

Understanding how to identify and use vertex coordinates is important in using the determinant formula effectively. In a 2D coordinate system, points are defined by their coordinates. These are ordered pairs \((x, y)\) where x is the horizontal position and y is the vertical position.

For instance, in our example, we have three vertices: \(\text{O}(0,0)\), \(\text{A}(4,2)\), and \(\text{B}(1,5)\). Here's how to identify them:

• \(\text{O}\): This is the origin, where both x and y are zero.
• \(\text{A}\): The point is 4 units to the right (x=4) and 2 units up (y=2).
• \(\text{B}\): The point is 1 unit to the right (x=1) and 5 units up (y=5).

With the coordinates clearly identified, you can easily substitute them into the determinant formula to find the area of the triangle.

Now let's calculate the area of the triangle using the provided coordinates. Recall vertices: \(\text{O}(0,0)\), \(\text{A}(4,2)\), and \(\text{B}(1,5)\).

Insert these into our formula: \[ \text{Area} = \frac{1}{2} \big| 0(2-5) + 4(5-0) + 1(0-2) \big| \]

Step-by-step:

• First term: \0 \times (2-5) = 0\
• Second term: \4 \times (5-0) = 20\
• Third term: \1 \times (0-2) = -2\

Combine these inside the absolute value: \[ \text{Area} = \frac{1}{2} \big| 0 + 20 - 2 \big| = \frac{1}{2} \big| 18 \big| = \frac{1}{2} \times 18 = 9 \text{ square units} \]

Therefore, the area of the triangle is 9 square units. Using the determinant formula can simplify your calculations for any triangle, making it a valuable tool in geometry.