Gradient and length

The gradient, often denoted as 'm', is calculated as the change in y divided by the change in x. The formula is: m = (y₂ - y₁) / (x₂ - x₁).

'Rise over run' is a visual representation of gradient. 'Rise' refers to the vertical change (change in y), and 'run' refers to the horizontal change (change in x). The gradient is the ratio of the rise to the run.

Use the distance formula (Pythagoras): √[(x₂ - x₁)² + (y₂ - y₁)²]. So, length AB = √[(5-2)² + (7-3)²] = √(9 + 16) = √25 = 5.

The distance between two points is the hypotenuse of a right-angled triangle. The legs of the triangle are the difference in x-coordinates and the difference in y-coordinates. Thus, a² + b² = c² (distance squared).

The midpoint is found by averaging the x-coordinates and averaging the y-coordinates. Midpoint = [(x₁ + x₂) / 2, (y₁ + y₂) / 2]. Therefore, the midpoint is [(1+5)/2, (4+2)/2] = (3, 3).

A negative gradient indicates that the line slopes downwards from left to right. As the x-value increases, the y-value decreases.

Using the distance formula: 10 = √[(7-1)² + (y-1)²]. Solving for y: 100 = 36 + (y-1)², so (y-1)² = 64. Thus y-1 = ±8, giving y = 9 or y = -7.

A gradient of zero indicates that the line is horizontal. The y-value remains constant, regardless of the x-value.

An undefined gradient means the line is vertical. This occurs when the change in x is zero (division by zero). The x-value remains constant, regardless of the y-value.