Coordinate Geometry Summary

Geometry · Axiom Academy

SUMMARY Unit 9 Summary: Coordinate Geometry Geometry meets algebra — a grid turns points into numbers, so distance, slope, and shape all become things you can compute . The coordinate plane pins every point to an ordered pair (x, y) ; the two axes split it into four quadrants with predictable sign patterns. Slope measures steepness and direction — positive, negative, zero (horizontal), or undefined (vertical). Lines are parallel when their slopes are equal ( m_1 = m_2 ) and perpendicular when their slopes are negative reciprocals ( m_1 m_2 = -1 ). The distance and midpoint formulas are just the Pythagorean theorem and averaging dressed up in coordinates. Any polygon on the grid yields its side lengths (distance), perimeter , and area (shoelace or decomposition) from its vertices alone. A coordinate proof places a figure with smart coordinates, then proves a property purely with distance, midpoint, and slope. Core Concept The Plane & Four Quadrants Each point is an ordered pair (x, y) : x runs left–right, y runs up–down. The axes cut the plane into four quadrants , numbered counter-clockwise from the top right. Sign map: I (+,+) , II (-,+) , III (-,-) , IV (+,-) . Watch out for: (x, y) order matters — . Slope is the constant rate of change of a line — how much y moves per step in x . Its sign tells you the line's tilt; its size tells you how steep. Positive rises L→R; negative falls; zero is horizontal. Undefined when x_2 = x_1 (vertical line — you'd divide by zero).

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