Problem

For segments \(AB\) and \(CD\) with \(A(0,0)\), \(B(2,6)\), \(C(1,4)\), \(D(4,3)\), calculate \(\text{both}~\text{slopes}\), apply the perpendicularity test, and state the conclusion.

Source coordinate plane: Segment AB runs from A(0,0) to B(2,6), and segment CD runs from C(1,4) to D(4,3); no slopes or perpendicular conclusion are shown.

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What this problem is really about

Compute both signed slopes before judging the drawing, especially preserving a negative rise when a segment falls from left to right. For nonvertical segments, multiply the slopes: a product of negative one verifies that they are negative reciprocals and therefore have perpendicular directions.

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mAB=3 and mCD=-1/3; test = 3(-1/3)=-1; conclusion = AB∥CD

mAB=-1/2 and mCD=2; test = (-1/2)(2)=-1; conclusion = AB⊥CD

mAB=(6-0)/(2-0)=6/2=3; mCD=(3-4)/(4-1)=-1/3; test = 3(-1/3)=-1; conclusion = AB⊥CD

mAB=3 and mCD=1/3; test = 3(1/3)=1; conclusion = AB⊥CD

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For segments \(AB\) and \(CD\) with \(A(0,0)\), \(B(2,6)\), \(C(1,4)\), \(D(4,3)\), calculate \(\text{both}~\text{slopes}\), apply the perpendicularity test, and state the conclusion.

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