Answer to Question #202039 in Geometry for Festus

Question

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

Answer 1

Given a quadrilateral $ABCD$ with $G$ and $H$ the midpoints of line $DA$ and $DC$ respectively. 

Consider triangle $ACD$

 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 $2GH=AC.$

Consider the square $ABCD$ with side $1$

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

Consider right triangle $ACD.$ By the Pythagorean Theorem

$$AC^2=AD^2+DC^2$$

$$AC=\sqrt{2}$$

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

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

Hence

$$2GH<AB+DC.$$

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