Answer to Question #202482 in Geometry for Festus

Question #202482

ABCD is a quadrilateral with G and H the mid points of line DA and DC respectively. Show that line AB + line DC = 2GH


1
Expert's answer
2021-06-08T04:31:20-0400


Given a quadrilateral ABCDABCD with GG and HH the midpoints of line DADA and DCDC respectively. 

Consider triangle ACDACD

 A midsegment of a triangle is a line segment which joins the midpoints of two sides of the triangle. It is parallel to the third side and has a length equal to half the length of the third side. Hence 2GH=AC.2GH=AC.


Consider the square ABCDABCD with side 11





AB=BC=CD=DA=1AB=BC=CD=DA=1

Consider right triangle ACD.ACD. By the Pythagorean Theorem


AC2=AD2+DC2AC^2=AD^2+DC^2

AC=2AC=\sqrt{2}

2GH=AC=22GH=AC=\sqrt{2}

AB+DC=1+1=2>2=2GHAB+DC=1+1=2>\sqrt{2}=2GH

Hence


2GH<AB+DC.2GH<AB+DC.


The statement  line AB + line DC = 2GH for a quadrilateral is False in our case.


In the example below 2GH>AB+DC2GH>AB+DC






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