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:

Glver: AABC
Prove: The midsegment between sides
AB and BC are parallel to side AC.
Draw AABC on the coordinate plane with point A at the origin (0,0). Let point B have the ordered pair (X1, Yı) and locate the point on the x-axis at
0+x 0+y

(X2, 0). Point D is the midpoint of AB, with coordinates at
by the slope formula. Point E is the midpoint of BC with an ordered
x+x. 0+y
pair by the slope formula. The slope of DE is found through the application of the slope formula:
[oty ory

0
=0
X + x 0+
Y, -Y 0-0 0
2 2
When the slope formula is applied to AC *-, X-OX,
Its slope is also O. Since
The slopes of DE and AC are identical, and DE and AC are parallel by the definition of parallel lines.
Which statement corrects the flaw in Gina's proof?
а
b
с
d
The coordinates of D and E were found using the midpoint formula.
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

Final Answer:

The correct statement to correct the flaw in Gina's proof is (a) "The coordinates of D and E were found using the midpoint formula."

Step-by-step explanation:

In Gina's original proof, the flaw lies in the incorrect statement that the slope of DE is found using the slope formula. The correct approach is to use the midpoint formula to find the coordinates of D and E. By locating the midpoints accurately, the proof becomes more robust.

The midpoint formula for a segment with endpoints (x₁, y₁) and (x₂, y₂) is given by:

(x₁ + x₂ / 2, y₁ + y₂ / 2)

In this context, using the midpoint formula ensures that the coordinates of D and E are determined correctly based on the midpoints of AB and BC. This correction strengthens the proof by providing accurate coordinates for the midpoints, which is crucial in geometrical reasoning.

Therefore, option (a) is the correct choice, stating that the coordinates of D and E were found using the midpoint formula. This correction addresses the flaw in the original proof and aligns with the proper procedure for establishing the midpoints in a geometric context.