When they say 'rectangular coordinate system', they just mean that the x-axis and y-axis make a right angle between them. So they're just talking about the standard x-y coordinate plane. It doesn't mean that any two points are equidistant from the origin. Statement 2 here is not sufficient, since you need some information about p.
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Similar questions to practice:
in-the-rectangular-coordinate-system-are-the-points-a-79105-20.html
in-the-rectangular-coordinate-system-are-the-points-r-s-92823.html
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My Answe:

We need information about all points to know if they are equidistant from (0,0)

A and B are eliminated due to not enough Info

The B) Trap is that we carry the info from A over.

C gives us |p| = |q| = |r| = |s|

When x<0 we have all values in Q1 and equal in distance
When x>0 we have all points in Q3 and equal in distance

Therefore C

Hi

Can someone please explain this?

Per the question, is distance of (p,q) from (0,0) equal to the distance of (r,s) from (0,0).

Distance of (p,q) from (0,0) = \(\sqrt{p^2+q^2}\), similarly for (r,s) = \(\sqrt{r^2+s^2}\)

Per statement 1, |p|=|q|, no information about (r,s). Clearly not sufficient.

Per statement 2, |q|=|r|=|s| , you get different answers if you have (p,q) , (r,s) = (0,3), (3,3) , the answer is no. But with (p,q) , (r,s) = (3,3), (3,3), the answer is yes. Thus you get 2 different answers for the same statement. Not sufficient.

Combining the 2 statements, you get, |p|=|q|=|r|=|s| and clearly for all cases you will get \(\sqrt{p^2+q^2}\) = \(\sqrt{r^2+s^2}\) ---> distance of (p,q) from (0,0) = distance of (r,s) from (0,0).

C is the correct answer.

Hope this helps.
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

In the rectangular coordinate system, are the points (p,q) and (r,s) equidistant from the origin?

(1) |p| = |q|

(2) |q| = |r| = |s|

We obtain square root [(p-0)^2+(q-0)^2]=square root[(r-0)^2+(s-0)^2], p^2+q^2=r^2+s^2? if we modify the question and the original condition.
There are 4 variables (p,q,r,s) but only 2 equations are given by the 2 conditions, so there is high chance (E) will become the answer.
Looking at the conditions together, from p^2=q^2=r^2=s^2 p^2+q^2=r^2+s^2? --> 2p^2=2p^2, we can answer the question 'yes' and the conditions become sufficient. The answer therefore becomes (C).

For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.
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Points that are equidistant are same in magnitude but vary in polarity. Any combination of the points (positive or negative) would be equidistant of they are all same in value. And the same goes with ordered pairs.
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