Problem

Verify that the ordered vertices \(P(0,0)\), \(Q(3,1)\), \(R(2,4)\), \(S(-1,3)\) form rectangle \(PQRS\). 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 PQRS has vertices P(0,0), Q(3,1), R(2,4), and S(-1,3); no slopes or rectangle conclusion are shown.

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side slopes = mPQ=1/3, mQR=-3, mRS=1/3, mSP=-3; parallel checks = PQ∥RS and QR∥SP; perpendicular check = (1/3)(-3)=3; conclusion = PQRS is not a rectangle

side slopes = mPQ=1/3, mQR=-3, mRS=1/3, mSP=-3; parallel checks = PQ∥RS and QR∥SP; perpendicular check = (1/3)(-3)=-1, so PQ⊥QR; conclusion = PQRS is a rectangle

side slopes = mPQ=1/3, mQR=-3, mRS=1/3, mSP=-3; parallel checks = PQ∥QR and RS∥SP; perpendicular check = (1/3)(-3)=-1; conclusion = PQRS is a rectangle

side slopes = mPQ=1/3, mQR=3, mRS=1/3, mSP=3; parallel checks = PQ∥RS and QR∥SP; perpendicular check = (1/3)(3)=-1; conclusion = PQRS is a rectangle

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