Foundations of Coordinate Geometry: Distance, Section, and Area The distance formula calculates the gap between two points by taking the square root of the sum of squared differences. Section formulas divide a line segment, with internal division yielding a weighted average and the midpoint representing an equal split, while external division modifies the rule with subtraction. The area of a triangle is determined using half the absolute value of a determinant formed by the coordinates.
Inclination and Slope: Measuring Line Orientation The inclination of a line defines its angle with the positive x-axis in an anticlockwise direction. Slope, computed as the tangent of this angle, quantifies the line’s steepness, remaining zero for horizontal lines and undefined for vertical lines. Parallel lines share the same slope, whereas the product of the slopes of perpendicular lines is -1.
Multiple Representations of Line Equations Lines can be expressed in various forms, including horizontal (y = constant), vertical (x = constant), slope-point, and two-point representations. The slope-intercept and intercept forms simplify the identification of slopes and axis intersections. Converting from a general equation to these forms involves isolating variables and normalizing constants, often resulting in the normal or distance form.
Determining Relative Positions and Intersections The position of points relative to a line is assessed by substituting their coordinates into the line’s equation and checking the sign of the result. The perpendicular distance from a point to a line is calculated using a formula that normalizes the linear combination of coordinates. Identifying the intersection of two lines or confirming the concurrency of multiple lines relies on solving their equations, often through determinants.
Perpendiculars, Reflections, and Families of Lines The foot of the perpendicular is found by aligning a point with the line using ratios that depend on the line’s coefficients, marking the closest approach between the point and the line. Reflecting a point across a line involves computing its image, with the foot of the perpendicular serving as the midpoint between the original and its reflection. Families of lines emerge by combining equations of intersecting lines or by determining lines through a fixed point that make a specified angle with a given line.
Angular Bisectors: Dividing Angles Precisely Angular bisectors split the angles formed by two intersecting lines into two equal parts. Their equations are derived using normalized forms of the line equations and include conditions that account for both possible bisectors. Determining whether a bisector corresponds to the acute or obtuse angle involves analyzing slope relations and sign criteria to ensure the correct orientation.