In the coordinate system above, the length of OS is 6. Which of the fo

Question

https://gmatclub.com/forum/download/file.php?id=47681 In the coordinate system above, the length of OS is 6. Which of the following must be true? I. The x-coordinate of point S is greater than -6. II. The slope of OS is greater than -1. III. The y-coordinate of point S is greater than -6. A. I only B. I and II only C. I and III only D. II and III only E. I, II, and III 2019-04-30_1207.png

IanStewart · Answer

Point S lies on a circle of radius 6, centered at the origin. The point on that circle with the smallest x-coordinate (i.e. the most negative x-coordinate) is on the x-axis, at (-6, 0). Every other point on that circle has a larger x-coordinate than -6, including the point S, so I must be true.

A line with slope exactly -1 (or exactly 1) will create a 45 degree angle with the x-axis. The line SO creates an angle less than 45 degrees, so its slope must be shallower than -1; it must have a slope between -1 and 0. If a number is between -1 and 0, then it is greater than -1, so II must be true.

The y-coordinate of point S is positive, so it is certainly greater than -6, so III must be true.

So all three of the items are true.

Sri07 · Answer

Bunuel

Could you please help understand conceptually and a bit in detail the point:
If the angle of the line with the X axis is 40 Degrees the slope would be greater than -1

Thanks in advance!

ShreyasJavahar · Answer

The line with slope 1 passes through the origin and points (1,1), (2.2), (3,3) in the first quadrant and (-1,-1),(-2,-2) and so on in the second, essentially y=x, and this line forms an angle of 45 degrees with both the x and y axes.
Similarly line with slope -1 passes through origin and points (-1,1), (-2,2) in first quadrant and (1,-1),(2,-2) in second, essentially y=-x, and this line also forms an angle of 45 degrees with the axes, you can plot this on a graph sheet and check.

So to obtain the answers to this question, simply compare the line with this relevant 45 degree line, y=-x in this case.
Naturally, a slope of 40 degree is lesser than a slope of 45 degrees, a road slanting 45 degrees is more steep than a road slanting 40 degrees.
But keep in mind that if -a < -b, then |a| > |b|, so line with slope 40 degrees is less negative and therefore greater than line with slope 45 degrees.

Fdambro294 · Answer

Using the Concept of drawing Right Triangles to use the Pythagoras Theorem to determine Distance/Length of a Slanted Line:

If we drew a Horizontal Line on the X-Axis from Origin (0 , 0) to the Point on the X-Axis that is Vertically Beneath Point S, we would create a Horizontal Leg of a Right Triangle. Since the Hypotenuse in this Right Triangle would = Line SO, which has a Length of 6, the Length of this Horizontal Leg would have to be LESS THAN< 6 Units.

Thus, the Distance ALONG the X-Axis from Origin (0 , 0) to the Point Directly, Vertically Beneath Point S must be LESS THAN< 6 Units.

The X Coordinate of S must therefore be Greater than -6 because the X Coordinate of Point S must be to the RIGHT of (-6 , 0) on the X-Axis.

I must be true.

The Line formed by the equation y = -x has a Slope = -1. One of the features of Line y= -x is that it creates a 45 degree Angle with the origin and the X-Axis when it Intersects through the Origin.

Starting from Slope =-1 of Line y= -x ---

as Lines with (-)Negative Slopes INCREASE IN VALUE and Approach 0, they will Approach the Horizontal Line with Slope = 0. The Lines will keep "Tilting Upwards" approaching the Horizontal Line with Slope = 0.

Therefore, since the Angle formed with the X-Axis by Line SO is 40 degrees and LESS THAN the 45 Degree Angle that Line y= -x forms with the X-Axis, Line Segment SO must be starting to "Tilt Upwards" and approach Slope = 0 as described above.

Therefore, because of this Fact, the Slope must be approaching 0 and be GREATER THAN > -1

II must be True

Lastly, III must be True. Point S is in Quadrant II in which all the Y Coordinates have (+)Positive Values

I, II, and III ALL must be true

-E-

In the coordinate system above, the length of OS is 6. Which of the fo

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In the coordinate system above, the length of OS is 6. Which of the fo

Prep Toolkit

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