Question #143558

Prove that the line joining the midpoints of the base and summit of a Saccheri Quadrilateral is the perpendicular bisector of both the base and the summit


1
Expert's answer
2020-11-11T12:29:34-0500

As we know in saccheri quadrilateral





As you see here AD=BC,DAB=CBA=90°AD=BC,\angle DAB=\angle CBA=90^{\degree}

    AB=CD,ABCD    ADC=BCD=90°\implies AB=CD,AB||CD\\\implies \angle ADC=\angle BCD=90^{\degree}


If we join the midpoints of base and summit, then then join

Let M and N be the midpoints of Summit and Base.


In ΔAMN\Delta AMN and ΔBMN\Delta BMN


AN=BNAN=BN ( as N is the midpoint of AB)

MN=MNMN=MN ( common side)

AM=BMAM=BM ( digonals of small quad. are equal)


ΔAMN+ΔBMN\therefore \Delta AMN + \Delta BMN


    BNM=ANM\implies \angle BNM=\angle ANM ( by C.P.C.T.)


As ANB is straight line

BNM+ANM=180°    2BNM=180°    BNM=90°\therefore \angle BNM+\angle ANM=180^{\degree}\\\implies 2\angle BNM=180^{\degree}\\\implies \angle BNM=90^{\degree}

So MNMN is the perpendicular bisector of Base ABAB as well as og summit CD.CD.


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