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JavyHart9695
JavyHart9695
12.11.2020 • 
Mathematics

A rectangle inscribed in a square is also a square. Proof Let rectangle MNPQ be inscribed in square ABCD as shown in Fig 2.6. Drop perpendiculars from P to AB and from Q to BC at R and S, respectively. Clearly PR  QS. Furthermore, PM  QN. SO PMR  QNS, and hence PMR  QNS. Consider quadrilateral MBNO where O is the point of intersection of QN and PM. Its exterior angle at the vertex N is congruent to the interior angle at the vertex M, so that the two interior angles at the vertices N and M are supplementary. Thus, the interior angles at the vertices B and O must also be supplementary. But ABC is a right angle, and hence, NOM must also be a right angle. Therefore the diagonals of rectangle MNPQ are perpendicular. Hence, MNPQ is a square.

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