If a and b are positive numbers, what are the coordinates of

Question

If a and b are positive numbers, what are the coordinates of the midpoint of line segment CD in the xy-plane?

(1) The coordinates of C are (a, 1-b) 
(2) The coordinates of D are (1-a, b)

Bunuel · Accepted Answer

Coordinates of the midpoint  M (x_m,y_m)  of the line segment AB, ( A (x_1,y_1)  and  B (x_2,y_2) ) are  x_m=\frac{x_1+x_2}{2}  and  y_m=\frac{y_1+y_2}{2}

(1) C (a, 1 - b). Not sufficient.

(2) D (1 - a, b). Not sufficient.

(1)+(2)  M (x_m,y_m)=M(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})=M(\frac{a+1-a}{2},\frac{1-b+b}{2})=M(\frac{1}{2},\frac{1}{2}) . Sufficient.

Answer: C.

If it were so, then answer would be E. In DS questions statement(s) is (are) sufficient only if we can get single numerical value of variable, midpoint coordinates, etc.

Ravshonbek · Answer

to find the mid point u just take the mean of x1 and x2 and mean of y1 and y2, that leaves us with mid point coordinates of x=1/2 y=1/2
to have it we need to use both statements
C

sandranjeim · Answer

So we find the coordinates of the midpoint in terms of a and b right?

Thanks

MathRevolution · Answer

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.

If a and b are positive numbers, what are the coordinates of the midpoint of line segment CD in the xy-plane?

(1) The coordinates of C are (a, 1-b) 
(2) The coordinates of D are (1-a, b)
 
There are 2 variables (x,y) and 2 equations are given by the 2 conditions, so there is high chance (C) will be the answer.
Looking at the conditions together,
midpoint=(a+1-a/2, 1-b+b/2)=(1/2,1/2)
This is unique and sufficient,
so the answer becomes (C).

For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% 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 C 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, D or E.

mcelroytutoring · Answer

I know what you mean (solving in terms of a variable is rarely sufficient), but we have to be careful--this is not always true.  If the question being asked is a yes/no question  ("is x positive?" or "is y less than 0?"), then the statements could be sufficient even if we can get multiple values for x or y, so long as they always result in the same answer to the question being asked (either "always yes", or "always no").

There are some cases on DS where both solving in terms of a variable, and getting multiple numerical values, can be sufficient.

For example, if the question asks, "is x > 5?" and statement #1 leads me to the conclusion that x = 10, 15, or 20, then I can conclude "sufficient," even though I found multiple solutions for x,  because the answer to the question being asked is always yes .

This can also be true when solving in terms of a variable. If the question asked "is x positive?" and I am able to solve for x as  (y^2+1) , then I know that regardless of the value of y, x must be positive. Hence it would be sufficient.

Bunuel · Answer

Yes. This is a value question so I think it's clear what I meant: When a DS question asks about the value of some variable, then the statement(s) is sufficient ONLY if you can get the  single numerical value  of this variable.

Hoozan · Answer

Similar (tad tricky): https://gmatclub.com/forum/what-are-the ... 43366.html

bumpbot · Answer

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If a and b are positive numbers, what are the coordinates of

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If a and b are positive numbers, what are the coordinates of

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