M05-13

Question

In xy plane, line K passes through the points A(6, -7) and B(4, 5). Does line K also pass through point C?

(1) Coordinates of Point C are ( 5, -1)

(2) Point C is equidistant from Point A and Point B.

Bunuel · Accepted Answer

Official Solution:

Notice that since we have two distinct points of line K, then we can find its equation.
 
(1) Coordinates of Point C are (5, -1). We know the equation of line K, hence we can find whether it passes through some particular point. Sufficient.
 
(2) Point C is equidistant from point A and point B. Point C may be the midpoint of the line segment AB, so on line K. But point C can also be anywhere on the  line which is perpendicular to line K and passes through that midpoint . Not sufficient.

Answer: A

BobsterGMAT · Answer

Bunuel  
Can you please tell me if we always assume that "the line" is straight?

Bunuel · Answer

Yes. In Euclidean geometry, a line is a straight curve. In coordinate geometry, lines in a Cartesian plane can be described algebraically by linear equations and linear functions.

Mo2men · Answer

Bunuel

Can you expain how the highlited part can be the case?

Thanks

Bunuel · Answer

Check the image below: https://gmatclub.com/forum/download/file.php?id=39242 Point (5, -1), red point, is the midpoint of (6, -7) and (4, 5). Red line is passing through that midpoint and is perpendicular to the blue line. Any point on the red line will be equidistant from (6, -7) and (4, 5). Untitled.png

tejyr · Answer

Hi Bunnel,
I might have a perpendicular line passing through C but C is on the line K right which in question we need to say whether k pass through C. In the above graph which you posted the point C is on line K. pls explain.

Bunuel · Answer

Any point on red line is equidistant from point A and point B, so C could be anywhere on that red line. If it's on the intersection (red dot) then it'll be on blue line to but if it's anywhere else on red line then it won't be on blue line.

tejyr · Answer

I was unable to understand how any point on the red line is equidistant from A and B.
For instance I took a point on the red perpendicular line (0,-11/6).
when i calculate the distance between (0,-11/6) and A
;(0,-11/6) and B are not equal.

Bunuel · Answer

The distance MUST be the same. The distance between (0, -11/6) and (4, 5) as well as the distance between (0, -11/6) and (6, -7) is  \frac{\sqrt{2257}}{6} .

I'll try to explain in another way. Consider points A = (0, 1) and B = (0, -1). Perpendicular bisector of AB is x-axis. Obviously any point on x-axis is equidistant from A and B.

The same way, any point on perpendicular bisector, of segment AB in our question, will be equidistant from A and B.

tejyr · Answer

My bad any point in red line is equidistant from A,b. Tq for explanation.

yogi02 · Answer

I was totally OK with the concept. I found that c is not on Line AB. Got the equation right, substituted Cs coordinates, and understood it doesn't pass.

But I failed to remember that it is Yes - No question and ans, in this case, is  No.

Friends keep this in mind. We often forget about it

UChisb · Answer

Bunuel  - can we canculate the slopes of AC and AB and compare if the slopes are equal?

Vubar · Answer

Sorry to rehash, but is it correct to say that any GMAT reference to a "line" is referring to a linear equation being plotted on the plane?

There is no possibility that they could be using the word line in the context of drawing the line representing a quadratic, for example?

Posted from my mobile device

Bunuel · Answer

In Euclidean geometry, a line is a  straight  curve. In coordinate geometry, lines in a Cartesian plane can be described algebraically  by linear equations and linear functions .

M05-13

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