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Two points, G and J with coordinates (c, d) and (m, n)
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04 Dec 2021, 23:48
04 Dec 2021, 23:48
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Question Stats:
28% (02:12) correct
71% (01:41) wrong based on 7 sessions
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For two points, G and J with coordinates (c, d) and (m, n), respectively, such that c < m, which of the following may be true for the line segment \(\bar{GJ}\) to have a positive slope?
A. c < m < 0
B. d > n > 0
C. n > d > 0
D. n = 0
E. d < 0, n > 0
A. c < m < 0
B. d > n > 0
C. n > d > 0
D. n = 0
E. d < 0, n > 0
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Official Answer
A,C,D,E
Re: Two points, G and J with coordinates (c, d) and (m, n)
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07 Dec 2021, 04:17
07 Dec 2021, 04:17
d can't be possible a, c, e are the only right Choices for this question
Posted from my mobile device
Posted from my mobile device
Two points, G and J with coordinates (c, d) and (m, n)
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08 Dec 2021, 23:35
08 Dec 2021, 23:35
1
Expert Reply
Solution:
Let's explore each option on an individual basis here:
Option A)
c < m < 0
Let's take the point G as (c, d) implies (-5, 0) and the point J as (m, n) implies (-4, 2), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{\Delta y}{\Delta x}\)
Slope of line \(\overline{GJ}\) = \(\frac{{2-0}}{{-4-(-5)}}\) = 2
Therefore, this option may be true.
Option B)
d > n > 0
Let's take the point G as (c, d) implies (-5, 3) and the point J as (m, n) implies (-4, 1), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)
Slope of line \(\overline{GJ}\) = \(\frac{{3 - 1}}{{-5 - (-4)}}\) = -2
Similarly, if we take the point G as (c, d) implies (5, 10) and the point J as (m, n) implies (7, 8), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)
Slope of line \(\overline{GJ}\) = \(\frac{{10 - 8}}{{5 - 7}}\)= -1
That's why, this option can not be true.
Option C)
n > d > 0
The points G and J can fall in the 1st or 2nd quadrant. Let's take the point G as (c, d) implies (-6, 7) and the point J as (m, n) implies (-4, 9), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)
Slope of line \(\overline{GJ}\) = \(\frac{{9 - 7}}{{-4 - (-6)}}\) = 2
Therefore, this option can be true.
Option D)
n = 0
We can assume the point G as (c, d) implies (-6, 6) and the point J as (m, n) implies (-4, 0), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)
Slope of line \(\overline{GJ}\) = \(\frac{{6 - 0}}{{-6 - (-4)}}\) = -3
Similarly, if we take the point G as (c, d) implies (-6, -6) and the point J as (m, n) implies (-4, 0), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)
Slope of line \(\overline{GJ}\) = \(\frac{{-6 - 0}}{{-6 - (-4)}}\)= 3
Therefore, this option can be true.
Option E)
d < 0 < n
Let's take the point G as (c, d) implies (-5, -2) and the point J as (m, n) implies (-3, 2), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)
Slope of line \(\overline{GJ}\) = \(\frac{{2 - (-2)}}{{-3 - (-5)}}\) = 2
That's why, this option may be true as well.
Hence, the correct answers are Option A), C), D) and E).
Let's explore each option on an individual basis here:
Option A)
c < m < 0
Let's take the point G as (c, d) implies (-5, 0) and the point J as (m, n) implies (-4, 2), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{\Delta y}{\Delta x}\)
Slope of line \(\overline{GJ}\) = \(\frac{{2-0}}{{-4-(-5)}}\) = 2
Therefore, this option may be true.
Option B)
d > n > 0
Let's take the point G as (c, d) implies (-5, 3) and the point J as (m, n) implies (-4, 1), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)
Slope of line \(\overline{GJ}\) = \(\frac{{3 - 1}}{{-5 - (-4)}}\) = -2
Similarly, if we take the point G as (c, d) implies (5, 10) and the point J as (m, n) implies (7, 8), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)
Slope of line \(\overline{GJ}\) = \(\frac{{10 - 8}}{{5 - 7}}\)= -1
That's why, this option can not be true.
Option C)
n > d > 0
The points G and J can fall in the 1st or 2nd quadrant. Let's take the point G as (c, d) implies (-6, 7) and the point J as (m, n) implies (-4, 9), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)
Slope of line \(\overline{GJ}\) = \(\frac{{9 - 7}}{{-4 - (-6)}}\) = 2
Therefore, this option can be true.
Option D)
n = 0
We can assume the point G as (c, d) implies (-6, 6) and the point J as (m, n) implies (-4, 0), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)
Slope of line \(\overline{GJ}\) = \(\frac{{6 - 0}}{{-6 - (-4)}}\) = -3
Similarly, if we take the point G as (c, d) implies (-6, -6) and the point J as (m, n) implies (-4, 0), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)
Slope of line \(\overline{GJ}\) = \(\frac{{-6 - 0}}{{-6 - (-4)}}\)= 3
Therefore, this option can be true.
Option E)
d < 0 < n
Let's take the point G as (c, d) implies (-5, -2) and the point J as (m, n) implies (-3, 2), then the slope will be:
Slope of line \(\overline{GJ}\) = \(\frac{{\Delta y}}{{\Delta x}}\)
Slope of line \(\overline{GJ}\) = \(\frac{{2 - (-2)}}{{-3 - (-5)}}\) = 2
That's why, this option may be true as well.
Hence, the correct answers are Option A), C), D) and E).
