Question

quadrilaterals wxyz and badc are congruent. in addition, \\( \overline { wx } \cong \overline { dc } \\) and \\( \overline { xy } \cong \overline { bc } \\). if \\( a d = 4 \mathrm { cm } \\) and \\( a b = 6 \mathrm { cm } \\), what is the perimeter of wxyz? 18 cm 20 cm 22 cm 24 cm

Explanation:

Step1: Use the property of congruent quadrilaterals

Since quadrilaterals \(WXYZ\) and \(BADC\) are congruent, their corresponding sides are equal. So \(WX = DC\), \(XY=BC\), \(YZ = AD\), \(ZW=AB\).

Step2: Find the lengths of the sides of \(WXYZ\)

Given \(AD = 4\mathrm{cm}\) and \(AB = 6\mathrm{cm}\). Then \(YZ=AD = 4\mathrm{cm}\), \(ZW = AB=6\mathrm{cm}\). Also, because \(WX\cong DC\) and \(XY\cong BC\), in quadrilateral \(BADC\), \(DC\) and \(BC\) are sides. Since \(BADC\) is a quadrilateral (assuming it is a parallelogram - but congruence gives the side - to - side relation directly), the perimeter of \(WXYZ\) is \(P=WX + XY+YZ + ZW\). And \(P=(WX + XY)+(YZ + ZW)\). Since \(WX = DC\) and \(XY = BC\), \(WX+XY=DC + BC\). In quadrilateral \(BADC\), \(DC + BC+AD + AB\) is its perimeter. But for \(WXYZ\), \(P=(WX + XY)+(YZ + ZW)=(DC + BC)+(AD + AB)\). Since \(WXYZ\cong BADC\), \(P = 2(AB + AD)\) (because \(AB = ZW\), \(AD=YZ\), \(WX + XY=BC + DC\) and in a quadrilateral \(BADC\), \(BC + DC=AB + AD\) if we consider the congruence and side - matching).

Step3: Calculate the perimeter

Substitute \(AB = 6\mathrm{cm}\) and \(AD = 4\mathrm{cm}\) into the formula \(P = 2(AB + AD)\).

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Answer:

\(20\mathrm{cm}\)