In the rectangular coordinate system, are the points (a, b)

[Anne1276]

* I don't understand the last piece of logic:
o If we know the proportion of a to b is the same as c to d and that |a| + |b| = |c| + |d|, then it must be the case that |a| = |c| and |b| = |d|. WHY IS THIS TRUE???

[Anne1276]

In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d

(2) square root of a squared + square root of b squared= square root of c squared +square root of d squared

[StaceyKoprince]

Hi, please remember to cite the name of the company that produced this test (even if it's us!).

[Anne1276]

Oops! MGMAT CAT #1 question 28. THANKS

[StaceyKoprince]

If we know the proportion of a to b is the same as c to d and that |a| + |b| = |c| + |d|, then it must be the case that |a| = |c| and |b| = |d|.

Given:
a/b = c/d
|a| + |b| = |c| + |d|

The mathematical proof for this is not short; I recommend learning this just as a "rule" the same way you know other math formulas and rules - it's not worth the time to derive and truly understand the proof. The first equation can also be written (using cross-multiplication) as bc = ad. Let's examine this using some random numbers.

b=2, c=6, a=3, d=4. Together, these give us bc=ad or 12=12. But they won't work in the equation |a| + |b| = |c| + |d| or 3 + 2 = 6 + 4.

In order to have the relevant sets of numbers both multiply to equal each other in one configuration (bc and ad) AND add to equal each other in another configuration (a+b and c+d), we need to have a number pair that will allow us to have one of each number assigned to (bc) and also to (ad) and ALSO one of each number assigned to (a+b) and also to (c+d). This is a fancy way of saying |a| = |c| and |b| = |d|. For example:

b = 2, c = 6
a = 6, d = 2
Together, these give us bc=ad or 12=12 AND |a| + |b| = |c| + |d| or 6 + 2 = 6 + 2.

[Anne1276]

Ok, thanks. I will simply learn that rule and use it from here on out. Thanks!

[Sam]

If we visualize two points (a,b) and (c,d) in the coordinate plane

a/b = c/d => the rise/run is same => slope is same => if (a,b) is in I-Quadrant then (c,d) is in III-Quadrant
or if (a,b) is in II-Quadrant then (c,d) will be in IV-Quadrant
==> implies that (a,b) & (c,d) can be any two points on line passing through origin.

Now if we combine (1){all points on a line passing through origin} with (2) |a| + |b| = |c| + |d| we get only two points, of which one point is mirror image of the other on this line passing through origin.

Hope I did add to the confusion.

[JonathanSchneider]

Be careful, Sam. Statement one does NOT, on its own, tell us that the points are in opposite quadrants. Both points could be in the SAME quadrant as well. Take, for example, (3, 4) and (6, 8).

You can tell this via Number Properties as well. For a/b to be equal to c/d, then the quotients of these terms must have the same sign. In other words, if a/b is positive, then c/d must also be positive. How would a/b be positive? Only if a and b have the same sign. Which quadrants have x and y values with the same sign? I and III. Thus, the points could be in opposite quadrants, but they could be in the same quadrant as well.