# The Gradient: Understanding and Implementing

• The gradient of a line is effectively its ‘steepness’.
• The gradient (m) can be calculated using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line.
• The idea of gradient can be pictorially represented as ‘rise over run’.
• The negative gradient signifies that the line slopes downwards as it moves from the left to right.
• A line that goes straight across (horizontal) has a gradient of zero.

# Parallel and Perpendicular Lines: Gradients and Relationships

• Lines that are parallel share the same gradient.
• Lines that are perpendicular have slopes that multiply together to give -1. So if Line 1 has gradient m1 and Line 2 has gradient m2, for the lines to be perpendicular, m1 * m2 = -1.

# The Pythagorean Method: Distances Between Two Points

• The distance between two points can be calculated using Pythagoras’ theorem: √[(x_2-x_1)² + (y_2-y_1)²].
• This forms the hypotenuse of a right triangle, where the horizontal and vertical distances between the two points are the other two sides of the triangle.

# Ratios and Coordinate Determination: Finding Points on a Line

• Any point lying between two given points (x1, y1) and (x2, y2) on a line can be found by using a ratio formula: (x, y) = ((1-k)x1+ kx2 , (1-k)y1+ ky2).
• By adjusting the value of k (which is the ratio), we can adjust where the point is on the line.

# Line Equations: The Slope-Intercept Form

• The equation of a straight line can be written in the form y = mx + c, where m is the gradient and c is the y-intercept.
• The y-intercept is the point where the line intersects (crosses) the y-axis.
• The gradient can be found from the slope-intercept form of the equation by looking at the coefficient in front of the x variable.