Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
Prove a quadrilateral is a rectangle using coordinates
Problem
Verify that the ordered vertices \(A(0,0)\), \(B(4,0)\), \(C(4,3)\), \(D(0,3)\) form rectangle \(ABCD\). Give \(\text{all}~\text{four}~\text{side}~\text{slopes}\), \(\text{both}~\text{opposite}\text{-}\text{side}~\text{parallel}~\text{checks}\), an adjacent-side perpendicular check, and the conclusion.
Build the rectangle proof in two layers: first establish a parallelogram from both opposite-side direction pairs, then verify one right angle between adjacent sides. When a side is vertical, classify it directly instead of treating its undefined slope as a number; horizontal and vertical directions supply the perpendicular check without multiplication.
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