Hiep is writing a coordinate proof to show that the midsegment of a trapezoid is parallel to its bases. He starts by assigning coordinates as given, where RS¯¯¯¯¯ is the midsegment of trapezoid KLMN. am I correct?

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1 Answer
Sean
Mar 3, 2018

Please see below.

Explanation:

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If #(x_1,y_1)# and #(x_2,y_2)# are coordinates of the end points of a line segment the coordinates of the midpoint can be found using the following formulas:

#((x_1+x_2)/2,(y_1+y_2)/2)#

In order to prove that #RS# is the midsegment of the trapezoid, we need to prove that #R# is the midpoint of #KN#, and #S# is the midpoint of #LM# and #RS# is parallel to the bases.

The coordinates of #R# and #S# are:

#R((2b+0)/2,(2c+0)/2)=R(b,c)#

#S((2a+2d)/2,(2c+0)/2)=S(a+d,c)#

#Slope_(RS)=m=(y_2-y_1)/(x_2-x_1)=(c-c)/(a+d-b)=0/(a+d-b)=0#

#Slope_(NM)# and #Slope_(KL)# are also #0#

Since the all three slopes are equal, all three lines are parallel to each other.

Therefore, #RS# is midsegment of the trapezoid.

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