In the xy-plane, line K passes through the point (1,1) and line m passes through point (1,-1). Are lines K and m prependicular to each other?

(1) Lines K and m intersect at the point (1,-1)

(2) Line K intersect the x-axis at the point (1,0)

I am not sure how to even solve this probelm. i really do suck at xy-plane problems. I choose answer "D" and offcourse that is wrong. Can someone please help?

OA: E

My suggestion will be to quikly draw those two points and proceed.

If you look at question Line K passes through point (1,1) and M passes through (1,-1).
If you quickly draw those two points, One is in 1st quadrant and other in 4th. This is what the question stem tells you. And the question is are the two lines prependicular , mathematically : is the product of the slopes of line K & M = -1?

Statement 1:
Tells us that lines interesect at point 1,-1.
If you would have drawn the above two points and now you join (1,1) and (1,-1) , which are the two points of line K, you will know the line K.
Line K is parallel to Y axis.
This is it , nothing else is given.

Whereas, all we know about line M is that it passes through 1,-1. And there can be infinite such lines. We cant conclude whether the two lines are prependicular or not.

INSUFFICIENT.

Statement 2:
This tells us that Line K passes through 1,0. Now if you join 1,1 and 1,0 . Once again you will get a line parallel to Y axis. Again if you extend this line you will relaize that it will cross point 1,-1. But situation is similar to St.1.

INSUFFICIENT

Combining 2 gives us line k. Which we got from both statements as well.
So together INSUFFICIENT as well.

Wow – Such a great explanation. Thanks a lot Nitin

nitin_prakash_khanna's explanation and approach is good. His first suggestion is very important: Draw the picture. It's much easier to do when you have a visual of what's going on.

In order to determine whether the lines are perpendicular, we need to determine the slopes of the two lines. One way we can rephrase the question is: do we know the slopes of lines k and m?

In order to determine the slopes of each line, we need two points for each line. Another way we can rephrase the question is: do we have two unique points for each line? Note that the problem has one point given for each line; we just need another point for each line.

Statement (1) provides an additional point for line k, but not for line m. We can determine the slope for only line k, but not m. So therefore, this statement is insufficient.

Statement (2) also provides an additional point for line k, but not for line ml. We can determine the slope for only line k, but not m. So therefore, this statement is insufficient.

When combining statements (1) and (2), we now have THREE points for line k, but only ONE point for line m. We still are unable to determine the slope for line m, so combined, these statements are insufficient. The answer should be E.

_________________
Ben Ku
Instructor
ManhattanGMAT

Ben, i would like to know why it is necessary to have two unique points.

My understanding...
To determine whether two lines are perpendicular, we need to see if the product of the gradients equals -1,
say m1*m2 = -1

Now all we need to determine the gradient of any line is two points on the line.. where on the line those two points are doesn't matter.. all we need is two points

statement 1 provides us with the point of intersection, so now we have a point for each of the two lines... using the question stem, we will have two points for each of the lines and therefore be able to obtain the gradients for each of the lines. Thus, we will be able to determine whether or not the two lines are perpendicular using the data from statement 1
Note: We don't need to perform any calculations, since we just have to determine sufficieny

I don't see why we need two unique points for each line... could you please explain why this isn't the case?

Ben thank you very much for your detail explination. it was really helpfull in getting my conecpts right.

For those who may be following along: the gradient is just another word for slope.

You're right. All we need are two (different) points to determine the slope of a line.



In statement (1), we can determine that line K passes through points (1, 1) and (1, -1), so we can find the slope of line K.

However, for line m, we only know it passes through (1, -1) because the given point and the point of intersection are the same. We don't have a second point and cannot determine the slope of line m.

Therefore, (1) is insufficient, because we only have the slope of line k but not line m. Hope that makes sense.

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Ben Ku
Instructor
ManhattanGMAT

Nice explanation Ben.

Nitin's approach was also quite good.

Aditya

Glad that Ben was able to help.