Kool problem - Geometry - MGMAT

[AleksandrM]

This one was fun. It was the last problem on my CAT, and I had just enough time to note the relationship between the slopes (hint):

Line l is defined by the equation y – 5x = 4 and line w is defined by the equation 10y + 2x + 20 = 0. If line k does not intersect line l, what is the degree measure of the angle formed by line k and line w?

A) 0
B) 30
C) 60
D) 90
E)It cannot be determined from the information given.

Enjoy!

[egybs]

That is a fun little problem..

we know the following:

l --> y = 5x + 4
w --> y = (-1/5)x + 2

Notice that the slopes are reciprocal... which means that they are perpendicular.

Now they introduce line K... which they tell us doesn't intersect L - this of course means that they are parallel. Since L and W are perpendicular, and since L and K are parallel, we know that W and K are also perpendicular.

So the answer is 90 degrees, D.

[anayak]

Slope of line l: 5 (Y=mX+b)
Slope of line 2: -1/5

Product of slopes = (-1)

that means two lines are perpendicular(90 degree).

[vittalgmat]

Hi,
I dint quite understand how u concluded that line K, if it does not intersect L, must be parallel. We dont know anything thing about K, such as its slope, etc. So I concluded that info is insufficient.

Pls explain.

thanks
-Vittal

That is a fun little problem..

we know the following:

l --> y = 5x + 4
w --> y = (-1/5)x + 2

Notice that the slopes are reciprocal... which means that they are perpendicular.

Now they introduce line K... which they tell us doesn't intersect L - this of course means that they are parallel. Since L and W are perpendicular, and since L and K are parallel, we know that W and K are also perpendicular.

So the answer is 90 degrees, D.

[egybs]

If there are two lines... regardless of their slopes or y intercepts, they will always intercept somewhere in the xy plane.... UNLESS they are parallel.. in which case they will never intersect. Since the question tells us they never intersect, we know they must be parallel.


All that said, you could probably argue the definition of "line" in this context.

I would guess that this isn't an official GMAT question, because it is a little unclear... unless I'm overthinking this and line ALWAYS means a line defined by y=mx+b.

Hi,
I dint quite understand how u concluded that line K, if it does not intersect L, must be parallel. We dont know anything thing about K, such as its slope, etc. So I concluded that info is insufficient.

Pls explain.

thanks
-Vittal

[Ian Stewart]

If there are two lines... regardless of their slopes or y intercepts, they will always intercept somewhere in the xy plane.... UNLESS they are parallel.. in which case they will never intersect. Since the question tells us they never intersect, we know they must be parallel.


All that said, you could probably argue the definition of "line" in this context.

I would guess that this isn't an official GMAT question, because it is a little unclear... unless I'm overthinking this and line ALWAYS means a line defined by y=mx+b.

The definitions of the lines seems all fine to me, but you're right that the question is badly worded. It says "what is the degree measure of the angle formed by line k and line w". If two lines intersect in a point, they will normally form two different angles, one larger than 90, one smaller than 90, except in the case where the lines meet at right angles. So the wording of the question essentially gives away the answer- it must be 90, or they'd have to be more specific about which angle they mean- the larger one or the smaller one.

[surbhi79]

That is a fun little problem..

we know the following:

l --> y = 5x + 4
w --> y = (-1/5)x + 2

The second line ... the equation given was 10y + 2x + 20 = 0 which means that the line is identified by:
w --> y = -x/5-2

How this work as a perpendicular in that case?

[chidcguy]

If there are two lines... regardless of their slopes or y intercepts, they will always intercept somewhere in the xy plane.... UNLESS they are parallel.. in which case they will never intersect. Since the question tells us they never intersect, we know they must be parallel.


All that said, you could probably argue the definition of "line" in this context.

I would guess that this isn't an official GMAT question, because it is a little unclear... unless I'm overthinking this and line ALWAYS means a line defined by y=mx+b.



We cant tell that those lines are parallel, just because they dont intersect. Even parallel lines intersect in infinity. Actually, there was another Q which MGMAT tested this. IMO, In coordinate geometry Q's we can only tell two lines are parallel if their slopes are equal.

[rey.fernandez]

I agree that the wording of the problem should be more precise. That said, I can only think of one situation where two lines do not intersect and yet they are not parallel: two skew lines. In this case, the two lines do not lie on the same plane.

In Euclidean geometry, parallel lines never intersect, provided they are on the same plane. In this problem, all of the lines are (implicitly) on the xy plane.

chidcguy, would you explain your statement that "even parallel lines intersect at infinity?"

[Ian Stewart]

We cant tell that those lines are parallel, just because they dont intersect. Even parallel lines intersect in infinity. Actually, there was another Q which MGMAT tested this.

The dangers of knowing too much math! There will never be a question on the GMAT about projective geometry- geometry on the GMAT is Euclidean only. That means that, on the GMAT, distinct parallel lines never intersect, and there are no 'points at infinity' to be concerned with. In more advanced math, geometries can be defined in all kinds of ways (by changing Euclid's axioms), and yes, in advanced geometry, distinct parallel lines can intersect, depending what kind of geometry you're working in. But not on the GMAT.

I'd be curious to see the MGMAT question you mention, incidentally; I'd be very surprised if it actually tested concepts from non-Euclidean geometry.

The discussion reminds me of the equation x^2 = -1. A mathematician would say this has two solutions, i and -i. On the GMAT, it has no solutions, of course.