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: ΔABC


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


Draw ΔABC 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). Point D is 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 Distance between Two Points Postulate. Point E is the midpoint of Line segment BC with coordinates of Ordered pair the quantity x sub 1 plus x sub 2 divided by 2. The quantity 0 plus y sub 1 divided by 2 by the Distance between Two Points Postulate. 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.


Which statement corrects the flaw in Gina's proof?


The slope of segments DE and AC is not 0.

Segments DE and AC are parallel by construction.

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

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

Answer 1

Answer:

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

Answer 2

Answer: The answers are

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

(ii) The coordinates of D and E were found using the Midpoint Formula.

Step-by-step explanation:  We can easily see in the proof  that the co-ordinates of D and E were found using the mid-point formula, not distance between two points formula. So, this is the first flaw in the Gina's proof.

Also, we see that the slope of line DE and AC, both are same, not equal to 0 but is equal to

$\dfrac{y_2}{x_2},$

which is 0 only if $y_2=0.$

So, this is the second mistake.

Thus, the statements that corrects the flaw in Gina's proof are

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

(ii) The coordinates of D and E were found using the Midpoint Formula.