BTGmoderatorLU wrote:In the x-y plane, the coordinates of three consecutive vertices of a rectangle are (2a, b), (-2a, b), (-2a, -3b). If a and b are integers with a<0 y b >0. Which of the following could be the coordinates of the fourth vertex?
A. (-3, -2)
B. (-2, 3)
C. (6, -9)
D. (-6, -9)
E. (-6, 9)
The OA is D.
Both B and D are possible.
For B, a = -1, b = 1
(-2,1), (2,1), (2,-3), and (-2,-3) for the coordinate (2a, -3b)(a square is a rectangle)
For D, a = -3, b = 3
Are the answer choices correct? Or am I missing something really obvious? Can anyone assist me please? Thanks in advance!
The axes of a two-dimensional Cartesian system divide the XY plane into four infinite regions, called quadrants, each bounded by two half-axes .
These are often numbered from I, II, III, and IV (numbered anti-clockwise):
- I (where the signs of the (x, y) coordinates are (+, +)),
II (−,+),
III (−,−), and
IV (+,−).
We are given that the three vertices are: (2a, b), (-2a, b), and (-2a, -3b) such that a < 0 and b > 0.
Thus, we can rewrite the coordinates of the three vertices as (-2|a|, |b|), (2|a|, |b|), and (2|a|, -3|b|).
Since
- of (-2|a|, |b|), the x coordinate is negative and the y coordinate is positive, the vertice must lie in the II quadrant;
of (2|a|, |b|), the x, as well as y coordinate, is positive, the vertice must lie in the I quadrant;
of (2|a|, -3|b|), the x coordinate is positive and the y coordinate is negative, the vertice must lie in the IV quadrant
So, three of the four vertices lie in three different quadrants (I, II and IV), thus, the fourth vertice would lie in the III quadrant.
We know that a point that lies in the III quadrant has its x, as well as y coordinate negative, thus, out of the five options, A or D can be the correct answer.
The coordinates of the fourth vertice would be (-2|a|, -3|b|). We are given that a and b are integers, thus, Option A cannot be correct answer as -2|a| ≠-3, else a would not be an integer.
Thus, the correct answer is D.
Let's check if a and b are integers.
We have -2|a| = -6 => |a| = 3, an integer; similarly, -3|b| = -9 => |b| = 3, an integer.
The correct answer:
D
Hope this helps!
-Jay
_________________
Manhattan Review GMAT Prep
Locations:
Manhattan Review India |
Hyderabad GMAT Coaching |
Bangalore GMAT Courses |
Visakhapatnam | and many more...
Schedule your free consultation with an experienced GMAT Prep Advisor!
Click here.
