Area of triangles

Here are the key points to learn when studying the area of a triangle:

1. Basic Formula: The area $A$ of a triangle can be calculated using the formula:

   $$A = \frac{1}{2} \times \text{base} \times \text{height}$$

2. Base and Height: The base can be any side of the triangle. The height is the perpendicular distance from the base to the opposite vertex.

3. Heron's Formula: For triangles where the height is not easily determined, the area can be computed using Heron's formula:

   $$A = \sqrt{s(s-a)(s-b)(s-c)}$$

   where $s = \frac{a+b+c}{2}$ is the semi-perimeter, and $a, b, c$ are the lengths of the sides.

4. Special Cases:

   • For right triangles, the two legs can serve as the base and height.
   • For equilateral triangles, the area can be calculated as:
   $$A = \frac{\sqrt{3}}{4} a^2$$

   where $a$ is the length of a side.

5. Coordinate Geometry: The area can also be calculated using the coordinates of the vertices:

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

6. Scale Factor: If a triangle is scaled by a factor $k$, the area is scaled by $k^2$.

7. Units: The area is expressed in square units corresponding to the units used for the base and height measurements.

These points provide a foundational understanding of how to calculate and conceptualize the area of a triangle in various contexts.