Algebra and geometry merge on the coordinate plane. Distance, midpoint, slope, and equations of lines let us prove geometric facts analytically.
Distance between (x₁,y₁) and (x₂,y₂): d = √[(x₂−x₁)²+(y₂−y₁)²]. Derived from the Pythagorean Theorem on the coordinate plane.
Midpoint M = ((x₁+x₂)/2, (y₁+y₂)/2). The average of the coordinates.
m = (y₂−y₁)/(x₂−x₁). Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals: m₁×m₂ = −1.
A circle centered at (h,k) with radius r: (x−h)² + (y−k)² = r². Expanding gives the general form.
Q1: Distance between (1,2) and (4,6):
Q2: The midpoint of (2,−4) and (8,10) is:
Q3: Two perpendicular lines have slopes 3 and: