5.4 Length and Area

- Use the **distance formula** between two points A(x₁, y₁) and B(x₂, y₂):

→ **Length = √[(x₂ - x₁)² + (y₂ - y₁)²]**

- This comes directly from Pythagoras’ Theorem.

- Example: Distance between (1, 2) and (4, 6):

→ √[(4 - 1)² + (6 - 2)²] = √[9 + 16] = √25 = 5

- If you know the coordinates of all 3 vertices, you can use:

→ **Area = ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|**

- This formula avoids needing to calculate base/height separately.

- Example: Triangle with vertices (1, 1), (4, 1), and (1, 5):

→ Area = ½ |1(1 - 5) + 4(5 - 1) + 1(1 - 1)| = ½ |-4 + 16 + 0| = ½ × 12 = 6

- A **trapezium** has two parallel sides. In coordinate geometry, find the lengths of the parallel sides and the height between them.

- Formula: **Area = ½ × (a + b) × h**, where:

- **a** and **b** are the lengths of the two parallel sides

- **h** is the perpendicular distance between them

- Use the gradient to check if two sides are parallel (same gradient).

- Use the distance formula to find side lengths and height.

- These methods apply to real-world shapes like land plots, ramps, signs, etc.

- Common problems include:

- Finding the area enclosed by lines and the axes

- Working out if triangles are isosceles or right-angled

- Verifying that lines form rectangles or trapezia