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 Midpoint Formula. 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 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 Parallel Postulate. Which statement corrects the flaw in Gina's proof? The coordinates of D and E were found using the slope formula.
Segments DE and AC are parallel by definition of parallel lines.
The coordinates of D and E were found using the Distance between Two Points Postulate. The slope of segments DE and AC is not 0.
Segments DE and AC are parallel by definition of parallel lines.
The coordinates of D and E were found using the Distance between Two Points Postulate. The slope of segments DE and AC is not 0.
1 answer:
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A. Factor the numerator as a difference of squares:
c. As
, the contribution of the terms of degree less than 2 becomes negligible, which means we can write
e. Let's first rewrite the root terms with rational exponents:
![\displaystyle\lim_{x\to1}\frac{\sqrt[3]x-x}{\sqrt x-x}=\lim_{x\to1}\frac{x^{1/3}-x}{x^{1/2}-x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bx%5Cto1%7D%5Cfrac%7B%5Csqrt%5B3%5Dx-x%7D%7B%5Csqrt%20x-x%7D%3D%5Clim_%7Bx%5Cto1%7D%5Cfrac%7Bx%5E%7B1%2F3%7D-x%7D%7Bx%5E%7B1%2F2%7D-x%7D)
Next we rationalize the numerator and denominator. We do so by recalling
In particular,
so we have
For
and 
, we can simplify the first term:
So our limit becomes
c. As
e. Let's first rewrite the root terms with rational exponents:
Next we rationalize the numerator and denominator. We do so by recalling
In particular,
so we have
For
So our limit becomes