Proofs involving quadrilaterals II

Learning Objectives Prove properties of quadrilaterals using coordinate geometry (slope, distance, and midpoint formulas). Prove that the diagonals of specific quadrilaterals have certain properties (e.g., bisect each other, are perpendicular, are congruent). Use triangle congruence postulates (SSS, SAS, ASA, AAS) within multi-step quadrilateral proofs. Classify a quadrilateral on the coordinate plane as a parallelogram, rectangle, rhombus, or square. Construct two-column proofs to justify statements about quadrilaterals. Prove that a quadrilateral is a specific type based on the properties of its diagonals. Ever wondered how a GPS knows the most direct route or how architects design perfectly symmetrical buildings? 🗺️ It all comes down to the powerful geometry of shapes! I...

TermDefinitionExample Coordinate ProofA proof that uses figures in the coordinate plane and algebra to prove geometric concepts. It bridges the gap between geometry and algebra.To prove a square's diagonals are equal, place its vertices at A(0, 0), B(s, 0), C(s, s), and D(0, s) and use the distance formula on AC and BD. Slope FormulaA measure of the steepness of a line, calculated as the change in vertical position (rise) divided by the change in horizontal position (run).The slope of a line through (2, 1) and (6, 9) is (9-1)/(6-2) = 8/4 = 2. Distance FormulaA formula derived from the Pythagorean theorem to find the straight-line distance between two points in a coordinate plane.The distance between (1, 2) and (4, 6) is √((4-1)² + (6-2)²) = √(3² + 4²) = √25 = 5 units. Midpoint Formul...

Slope Formula m = (y₂ - y₁) / (x₂ - x₁) Use to prove lines are parallel (slopes are equal) or perpendicular (slopes are negative reciprocals, where m₁ * m₂ = -1). Distance Formula d = √((x₂ - x₁)² + (y₂ - y₁)²) Use to prove segments are congruent (have equal length). Essential for checking side lengths and diagonal lengths. Midpoint Formula M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2) Use to prove that diagonals bisect each other. If two diagonals share the same midpoint, they cut each other in half.