OG In the xy-plane, the Origin is the midpoint

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AbeNeedsAnswers: Master | Next Rank: 500 Posts; Posts: 394; Joined: Sun Jul 02, 2017 10:59 amFollowed by: 5 members; Thanked: 1 times

by AbeNeedsAnswers » Thu Aug 17, 2017 7:16 pm

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In the xy-plane, the origin O is the midpoint of line segment PQ. If the coordinates of P are (r,s), what are the coordinates of Q ?

A. (r,s)
B. (s,-r)
C. (-s,-r)
D. (-r,s)
E. (-r,-s)

E

Source: Beat The GMAT — Problem Solving | Thu Aug 17, 2017 7:16 pm

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by Matt@VeritasPrep » Fri Aug 18, 2017 1:47 pm

If we've got two points (a, b) and (c, d), then their midpoint = ((a + b)/2, (c+d)/2).

Since the origin has the coordinates (0, 0), we need (a + b) = 0 and (c + d) = 0. This is possible with (r + -r) and (s + -s), so our missing coordinate is (-r, -s).

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by [email protected] » Sun Aug 27, 2017 1:36 pm

Hi AbeNeedsAnswers,

We're told that the origin O is the MIDPOINT of line segment PQ and that the coordinates of P are (r,s). We're asked for the coordinates of Q. This question can be solved by TESTing VALUES.

IF... Point P = (r,s) = (2,1)...
then Point Q would be (-2, -1). That equates to (-r,-s).

Final Answer: E

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by Admin1 » Sun Sep 03, 2017 9:07 am

If we have two points A=(x1,y1) and B=(x2,y2), then the midpoint of the line segment AB is

M= ( (x1+x2) / 2 , (y1+y2 )/ 2 ).

As the Origin is the Midpoint fo the line segment PQ, we have that M= (0,0).

Let's say that A=P, i.e. x1=r and x2=s. We want to calculate the coordinates of Q=(x2,y2).

So, we have: (0,0) = ( (r+x2)/2 , (s+y2)/2 ) which implies (r+x2)/2 = 0 and (s+y2)/2 = 0.

This is equivalent to r+x2=0 and s+y2=0.

Finally, we get x2=-r and y2=-s, i.e Q=(-r,-s).

So, the correct option is E

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by Scott@TargetTestPrep » Wed Nov 15, 2017 3:55 pm

AbeNeedsAnswers wrote:In the xy-plane, the origin O is the midpoint of line segment PQ. If the coordinates of P are (r,s), what are the coordinates of Q ?

A. (r,s)
B. (s,-r)
C. (-s,-r)
D. (-r,s)
E. (-r,-s)

We can let the coordinates of Q be (x, y). Since the origin (0, 0) is the midpoint of PQ, we have:

(r + x)/2 = 0 and (s + y)/2 = 0

Multiplying each equation by 2, we have:

r + x = 0 and s + y = 0

x = -r and y = -s

Thus, the coordinates of Q are (-r, -s).

Answer: E

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