Chapter 3: Coordinate Geometry

The foundational material establishes methods for computing essential properties of line segments, including the midpoint formula that averages paired coordinates, the distance calculation derived from Pythagorean principles, and the gradient or slope measurement that quantifies steepness. Building on these basics, the chapter explores how gradients reveal the relative positioning of two distinct lines: parallel lines share identical gradients and never intersect, while perpendicular lines meet at right angles with gradients whose product equals negative one, making each the reciprocal opposite of the other. The representation of lines takes multiple algebraic forms depending on available information, from the slope-intercept arrangement to the point-slope configuration and general linear equations, with special attention to vertical lines that resist standard gradient analysis. The chapter then shifts focus to circular geometry, defining circles as loci of equidistant points from a central location and providing both the completed-square standard form and the expanded general form with explicit formulas for recovering the center and radius from algebraic coefficients. Key geometric properties of circles receive attention, particularly the right-angle property of semicircles, the perpendicular bisector property of chords, and the perpendicular relationship between tangent lines and radii at contact points. The final section addresses intersection problems by solving line and circle equations simultaneously, generating quadratic expressions whose discriminant serves as a decisive tool: positive values indicate two distinct intersection points, zero values identify tangency conditions where the line touches the circle at exactly one location, and negative values confirm complete separation between the geometric figures. Throughout, the chapter emphasizes connections between algebraic equations and geometric properties, enabling students to visualize abstract equations as concrete spatial relationships.