Problem

Verify that the ordered vertices \(J(1,2)\), \(K(5,4)\), \(L(3,8)\), \(M(-1,6)\) form rectangle \(JKLM\). 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.

Source coordinate plane: Quadrilateral JKLM has vertices J(1,2), K(5,4), L(3,8), and M(-1,6); no slopes or rectangle conclusion are shown.

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side slopes = mJK=1/2, mKL=-2, mLM=1/2, mMJ=-2; parallel checks = JK∥LM and KL∥MJ; perpendicular check = only diagonal JL²=40; conclusion = JKLM is a rectangle

side slopes = mJK=1/2, mKL=-2, mLM=1/2, mMJ=-2; parallel checks = JK∥LM and KL∥MJ; perpendicular check = (1/2)(-2)=-1, so JK⊥KL; conclusion = JKLM is a rectangle

side slopes = mJK=1/2, mKL=-2, mLM=1/2, mMJ=-2; parallel checks = JK∥LM and KL∥MJ; perpendicular check = (1/2)(-2)=1; conclusion = JKLM is not a rectangle

side slopes = mJK=1/2, mKL=2, mLM=1/2, mMJ=2; parallel checks = JK∥LM and KL∥MJ; perpendicular check = (1/2)(2)=-1; conclusion = JKLM is a rectangle

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