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

[shaw.s.li]

Hi.

I got the following question during my first CAT

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(a^2) + square root(b^2) = square root (c^2) + square root (d^2).

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.

My specific question is regarding the explanation that says:

(1) AND (2) SUFFICIENT: Together the statements are sufficient. Why? 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|.

Can someone explain by the part that is bolded is true? I don't understand this logic. is the proportion of a to b is the same as c to d, why is it that |a| = |c| and |b| = |d|. Why part of math logic am I missing?

[gokul_nair1984]

Consider the following,
(a,b) = (1,2) & (c,d) = (3,6) so not equidistant from origin.
or if
(a,b) = (-1,-2) & (c,d) = (1,2) then equidistant from origin.
not sufficient.

By Considering both statements, the only values that fit in are:
(a,b) = (1,2) & (c,d) = (1,2)
or
(a,b) = (-1,-2) & (c,d) = (1,2)
All other combinations will yield answers that do not satisfy both equations.

...Hence b is not sufficient. Just plug in numbers to contradict a particular case.

Regarding your query,

|a| + |b| = |c| + |d|, then it must be the case that |a| = |c| and |b| = |d|.

Let us again Plug in some numbers..
a=2,b=3,c=4,d=1

By substituting, we can see that this need not be true.

[shaw.s.li]

Thank you for the info, but that doesn't really address my question.

I know why 1 alone isn't enough. (as you suggested, plug in numbers)

I also know why 2 alone isn't enough (as you suggested again, plug in some numbers).

I'm trying to understand that if you take statement 1 and 2 together, by |a| = |c| and |b| = |d|. I'm sorry, but your explanation doesn't answer that question.

[gokul_nair1984]

St 1 gives us ad=bc
St 2 gives us a+b=c+d---multiply this with b
(ab)+(b)^2=(cb)+(db)

ad=cb, substitute in the above equ

(ab)+(b)^2=(ad)+(db)
b(a+b)=d(a+b)
b=d

since b=d
ad=bc
a=c

or...|a| = |c| and |b| = |d|.


The x and y coords of the two points are same hence they are equidistant.


-------------

PS: Statement 2 is sqrt(a)^2+sqrt(b)^2=sqrt(c)^2+sqrt(d)^2,
which can be rewritten as a+b=c+d

Kindly let me know if I have been able to answer your query..

[shaw.s.li]

Thanks. That helps.

[mschwrtz]

Good enough then?

[chitrangada.maitra]

Thats really neat, Gokul!

Number plugging wasn't working for me - this helps!

[gokul_nair1984]

Thank You :)

[mschwrtz]

Oh, one other point. As perhaps you already understood Gokul, statement 2 gives us |a|+|b|=|c|+|d|, not a+b=c+d. Your reasoning after that still works, though. Very nicely done.

Now, having said that, I've seen students have much more success testing values on this question than solving formally.

[as2764]

Oh, one other point. As perhaps you already understood Gokul, statement 2 gives us |a|+|b|=|c|+|d|, not a+b=c+d.

very imp point! since, sqrt(a^2) = |a|, and NOT a

in essence -- st2 says that sum of moduli are equal and not the totality.

[jnelson0612]

Thanks all. I concur with Michael's point that testing numbers is the way to go on this problem. It's amazing how many problems are quickly solved by testing numbers.

[shaw.s.li]

Oh, one other point. As perhaps you already understood Gokul, statement 2 gives us |a|+|b|=|c|+|d|, not a+b=c+d. Your reasoning after that still works, though. Very nicely done.

Now, having said that, I've seen students have much more success testing values on this question than solving formally.

As a question, if Gokul's statement 2 above is |a|+|b|=|c|+|d|, not a+b=c+d, how can the rest of the reasoning still work? How can he multiple by b on all sides?

St 1 gives us ad=bc
St 2 gives us a+b=c+d---multiply this with b
(ab)+(b)^2=(cb)+(db)

ad=cb, substitute in the above equ

(ab)+(b)^2=(ad)+(db)
b(a+b)=d(a+b)
b=d

since b=d
ad=bc
a=c

or...|a| = |c| and |b| = |d|.


The x and y coords of the two points are same hence they are equidistant.


-------------

PS: Statement 2 is sqrt(a)^2+sqrt(b)^2=sqrt(c)^2+sqrt(d)^2,
which can be rewritten as a+b=c+d

Kindly let me know if I have been able to answer your query..

[tim]

multiplying by b has nothing to do with whether the existing equation has absolute value signs. it is always legal to multiply any equation by anything as long as it's not zero..

[madhu.a.punjabi]

Since Gokul took the absolute values off of the variables, is it even possible to solve algebraically?

And if we plug in numbers, how would you go about explaining it?

I'm sorry I'm just still having trouble understanding the logic.

[jnelson0612]

Since Gokul took the absolute values off of the variables, is it even possible to solve algebraically?

And if we plug in numbers, how would you go about explaining it?

I'm sorry I'm just still having trouble understanding the logic.

It makes so much more sense to just test numbers on this one than to try to do algebra.

The way we test numbers on data sufficiency is to pick numbers that fit the statements; in other words, number that make the statements true. Start with statement #1. Notice that Gokul has picked two sets of numbers, both of which fit the criteria for statement 1. He then takes his numbers and tries to answer the question in the stem. One set of numbers gives a YES answer; the other gives a NO answer. This statement is insufficient.

If that makes sense, please read through his explanation one more time and tell us if you get stuck, and if so where specifically you are stuck. I hope it makes more sense now.