Question

"Gina wrote the following paragraph to prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side.
Given: ∆ABC

Prove: The midsegment between sides Line segment AB and Line segment BC is parallel to side Line segment AC.

Draw ∆ABC on the coordinate plane with point A at the origin (0, 0). Let point B have the ordered pair (x1, y1) and locate point C on the x-axis at (x2, 0). Label point D as the midpoint of Line segment AB with coordinates at Ordered pair the quantity 0 plus x sub 1, divided by 2; the quantity 0 plus y sub 1, divided by 2 by the Midpoint Formula. Label point E so it is the midpoint of Line segment BC with an ordered pair of Ordered pair the quantity of x sub 1 plus x sub 2 divided by 2; the quantity of 0 plus y sub 1 divided by 2 by the Midpoint Formula. The slope of Line segment DE is found to be 0 through the application of the slope formula: The difference of y sub 2 and y sub 1, divided by the difference of x sub 2 and x sub 1 is equal to the difference of the quantity 0 plus y sub 1, divided by 2, and the quantity 0 plus y sub 1, divided by 2, divided by the difference of the quantity x sub 1 plus x sub 2, divided by 2 and the quantity 0 plus x sub 1, divided by 2 is equal to 0 divided by the quantity x sub 2 divided by 2 is equal to 0 When the slope formula is applied to Line segment AC the difference between y sub 2 and y sub 1, divided by the difference of x sub 2 and x sub 1 is equal to the difference of 0 and 0, divided by the difference of x sub 2 and 0 is equal to 0 divided by x sub 2 is equal to 0, its slope is also 0. Since the slope of Line segment DE and Line segment AC are identical, Line segment DE and Line segment AC are parallel by the definition of parallel lines.

What is the flaw in Gina’s proof?

Points D and E must be constructed, not simply labeled, as midpoints.

Segments DE and AC are parallel by construction.

The slope of segments DE and AC is not 0.

The coordinates of D and E were found using the Distance between Two Points Postulate."

Answer 1

What is the flaw in Gina’s proof?
A) Points D and E must be constructed, not simply labeled, as midpoints.

B) Segments DE and AC are parallel by construction.  THIS IS THE FLAW. THE SEGMENTS WERE NOT DRAWN NOR PROPERLY IDENTIFIED.

C) The slope of segments DE and AC is not 0.

D) The coordinates of D and E were found using the Distance between Two Points Postulate 

Answer 2

 

It is a simple and fun geometrical problem, and it makes all sense until:

 

"The slope of Line segment DE is found to be 0 through the application of the slope formula:"

 

After that it gets all confusing etc.  The slope formula applied to DE is simply:

(difference between the y coordinates) divided by (difference of the x coordinates).

In this case, by construction, D and E have the same y coordinate equal to y1 / 2.

Therefore the slope is zero.

 

Using the same technique, you will find that the slope of segment AC is also zero (by construction obviously since point A is the origin (0,0) and point C is on the x-axis.

 

Therefore:

The slope of segments DE and AC is not 0. = INCORRECT

Segments DE and AC are parallel by construction. = CORRECT (they have the same slope)

The coordinates of D and E were found using the Midpoint Formula. = CORRECT

The coordinates of D and E were found using the slope formula. = INCORRECT

 Thanks Jean-Michel at www.wyzant.com

Jean Michel T. answered • 10/21/15

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