Math L3&4 Coordinate Geometry

Coordinate Geometry

Coordinate Plane and Points

• The coordinate plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).
• The intersection of the x and y axes is the origin, denoted as O.
• The axes divide the plane into four quadrants (I, II, III, IV).
• Points are defined by coordinates (x, y), where x is the horizontal distance from the origin and y is the vertical distance from the origin.
• In quadrant I, both x and y are positive (+, +).
• In quadrant II, x is negative and y is positive (-, +).
• In quadrant III, both x and y are negative (-, -).
• In quadrant IV, x is positive and y is negative (+, -).

Distance, Midpoint, and Gradient

• Distance Between Points: The distance d between points P(x1, y1) and Q(x2, y2) is given by: d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}.
• Midpoint of a Line: The midpoint M of a line segment joining points A(x1, y1) and B(x2, y2) is: M = (\frac{x1 + x2}{2}, \frac{y1 + y2}{2}).
• Gradient (Slope): The gradient m of a line segment joining points A(x1, y1) and B(x2, y2) is: m = \frac{y2 - y1}{x2 - x1}.
• Parallel lines have equal gradients: m1 = m2.
• Perpendicular lines have gradients that satisfy: m1m2 = -1.

Equation of a Straight Line

• Point-Slope Form: y - y1 = m(x - x1)
• Slope-Intercept Form: y = mx + c, where m is the slope and c is the y-intercept.
• Standard Form: ax + by = c
• Collinearity: Points P, Q, R are collinear if the gradient of PQ equals the gradient of QR equals the gradient of PR.

Systems of Linear Inequalities

• To graph inequalities, first plot the corresponding equation as a line (dotted for strict inequalities, solid for inclusive inequalities).
• Determine which side of the line to shade by testing a point (e.g., (0, 0)) in the inequality.
• If the point satisfies the inequality, shade that side; otherwise, shade the opposite side.
• For systems of inequalities, the solution is the region that satisfies all inequalities simultaneously.

Area of a Triangle

• The area A of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by:
  A = \frac{1}{2} |x1y2 + x2y3 + x3y1 - x2y1 - x3y2 - x1y3|
• If the area is zero, the points are collinear.