In the xy-coordinate system, what is the slope of the line

B

line: y=ax

1. the line has to pass through the middle point (O) of the segment PQ.
2. the coordinates of the middle point are: (1+(7-1)/2, 7+(11-7)/2) = (4,9)
3. 9=a*4 --> a=2.25

How can you assume that the line will pass through the mid point of PQ.
PQ may not be perpendicular to the line passing through origin.

PM ┴ y=ax, QN ┴ y=ax

PM=QN, QNO=PMO=90°

angle POM = angle QON
angle OPM = angle OQN

So, ∆POM identical with ∆QON and PO=OQ

What is the slope of the line that goes through the origin and is equidistant from the two points p(1,11) and q(7,7)?

A. 2
B. 2.25
C. 2.50
D. 2.75
E. 3

thnkxx beyondgmatscore.Also i was more interested in some theory concepts.

@AnkitK, please look here :

math-coordinate-geometry-87652.html

Most co-ordinate geometry questions become way easier the moment you draw them out... the extra seconds almost always provide a lot of value... Knowing some basic formulas helps and you should be very efficient in drawing lines from their equations, from their slope and a point, from two points etc...

First, get the middle coordinate between (1,11) and (7,7)...
x = 1 + (7-1)/2 = 4
y = 7 + (11-7)/2 = 9

Second, get the slope of (4,9) and (0,0). m = 0-9 / 0-4 = 9/4 = 2.25

Answer: B

Hi All.. I went through the explanation given by walker for vshaunak's query.. I still dont find how are we assuming that those points are on opposite sides of the line...What if both are in the same side of the line?

Hi Rahghunandan,


You are absolutely CORRECT about your doubt.

However you need to understand the following two points

Case 1) Either a line that is equidistant from the two points must be passing from the the gap between the two points. In this case every point on the line will be equidistant from each of the two given points.

OR

Case 2) The line that is equidistant must be parallel to the line joining the two points (7, 7) and (1, 11). But this case is applicable only when the perpendicular distance of the Line from point is discussed

But in the case 1) the line will have the positive slope and in case 2) the line will have a negative slope [Slope = (11-7) / (1-7) = - (2/3)] and here we have not been given any option of Negative slope. Also the questions doesn't mentioned anything about the perpendicular distance of line from point specifically.

Therefore, We will have to consider case 1 only and find the slope of the line that passes through the gap between two lines and is equidistant from two points.

I hope it clears your doubt!!!

Hi All,

There are two ways 2 points can be equidistant from a line:

Option1: The line passes through the mid point of the two given points.

In this case find the mid point : X coordinate: (1+7)/2 & Y coordinate: (7+11)/2
Coordinates of the mid point: (4,9)
Now find the slope of the line between the mid point and the origin: ((9-0)/(4-0)) = 2.25


Option 2: The line between the given point P & Q is parallel to a line passing through the origin.

Slope of the line PQ : ((11-7)/(7-1)) =4/6 = 2/3
Slope of parallel lines are the same, hence the slope of the line parallel to PQ will also be 2/3

I am not sure if this is an official question or not but my understanding is that if this were to appear in actual GMAT then question will also provide some information that rules out the option of the line being parallel. E.g : The two line do intersect at some point.

Let me know if you found this useful!

Regards,
Shradha

Indeed, this seems to be a fair point that you've pointed out. However, I guess the slope for the parallel line would be (-2/3).

This question is certainly much easier if you have a solid board with a plane on it

https://www.amazon.com/Manhattan-GMAT-S ... %2F+Marker

Anyways, if we draw out points P and Q on a plane and create a triangle (this step is not really necessary but may help certain test takers) then we can see that height of the triangle is 4 and the length 6. Let our imaginary point be "K"- the x coordinate for this point would exist halfway on the length of the triangle so since our length is 6 half of the length is 3 and 3 spots from the x value of point P, "1" would be 4. The same holds true for the Y value- the y value of our point "K" exists halfway on the length of the triangle- another to think of it is what is the median of the values 7 8 9 10 11- on this logic the y value is 9. Finally, when the question says that the imaginary line passes through the original this simply means that there is no y intercept for the line's equation- in other words any equation of the line would not have the plus or minus part ( example: y= 4x +2- this has a y intercept; y= 5x -2 - this has a y intercept). Knowing this you can simply translate the individual answer choices into equations and solve until you find an equation that satisfies the coordinates (4,9)- for example

Choice E
y= 3x
9= 3(4)
9 does not equal 3(4)

Choice B
9= 2.25x
9= 2 (1/4)x
9= 2(1/4) 4
9= (9/4) 4
9= 36/4

Therefore
"B"

Since the line is equidistant from P = (1, 11) and Q = (7, 7), it must pass through the midpoint between P = (1, 11) and Q = (7, 7). We can use the midpoint formula:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

Midpoint = ((1 + 7)/2, (11 + 7)/2)

Midpoint = (4, 9)

Since the line also passes through the origin, (0, 0), the slope is:

Slope = change in y/change in x

(9 - 0)/(4 - 0) = 9/4 = 2.25

Answer: B

Hi,
Please excuse my misunderstanding, if 2 coordinate points of line are given i.e. P and Q, cant we find slope with the help of slop formula? m= y2-y1/x2-x1?

This is a pretty simple question.

Asked: In the xy-coordinate system, what is the slope of the line that goes through the origin and is equidistant from the two points P = (1, 11) and Q = (7, 7)?

Coordinate of point equidistant from the two points P = (1, 11) and Q = (7, 7) = (4,9)

Slope of line passing through origin (0,0) and (4,9) = 9/4 = 2.25

IMO B