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In the xy-plane, the line y = k is the perpendicular bisector of the line segment PQ and the line x = h is the perpendicular bisector of the line segment RQ. If the coordinates of the point R are (-h, -k), then what are the coordinates of the point P?
- a)(-5h, -5k)
- b)(-3h, -3k)
- c)(2h, 2k)
- d)(3h, 3k)
- e)(5h, 5k)
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
In the xy-plane, the line y = k is the perpendicular bisector of the l...
Step 1: Question statement and Inferences
Drawing the axes and the lines y = k and x = h given in the question on xy-plane:
Based on the figure drawn by us, plotting the point R whose coordinates are (-h, -k): 
Step 2:Finding required values
Since the line x = h is the perpendicular to QR, the y-coordinate of point R must also be equal to –k. RA and QA are perpendicular to the line x = h, the y-coordinate of R, A and Q are equal. Therefore, the coordinates of point A are (h, -k).
RA = AQ = h – (-h) = 2h
The coordinates of point Q are (h + 2h, -k) or (3h, -k).
Step 3: Calculating the final answer
Since the line y = k is perpendicular to PQ, the x-coordinate of P, B and Q are equal, i.e. 3h.
Since QB = BP, and QB = k – (-k) = 2k, the y-coordinate of point P is k + 2k = 3k.
The coordinates of the point P are (3h, 3k).
(D) is the correct answer.
Most Upvoted Answer
In the xy-plane, the line y = k is the perpendicular bisector of the l...
To find the coordinates of point P, we need to understand the properties of perpendicular bisectors.
The Perpendicular Bisector Property:
A line is the perpendicular bisector of a line segment if it passes through the midpoint of the segment and is perpendicular to it.
Given Information:
1. The line y = k is the perpendicular bisector of the line segment PQ.
2. The line x = h is the perpendicular bisector of the line segment RQ.
3. Point R has coordinates (-h, -k).
Using the Perpendicular Bisector Property:
Since the line y = k is the perpendicular bisector of line segment PQ, the midpoint of PQ lies on the line y = k. Let the midpoint be M.
Since the line x = h is the perpendicular bisector of line segment RQ, the midpoint of RQ lies on the line x = h. Let the midpoint be N.
Now, let's find the coordinates of M and N:
1. Midpoint M:
Since the line y = k is the perpendicular bisector of PQ, M lies on the line y = k. Therefore, the y-coordinate of M is equal to k.
Since M is the midpoint of PQ, we can find its x-coordinate by taking the average of the x-coordinates of P and Q. Let's assume the coordinates of P are (x1, y1) and the coordinates of Q are (x2, y2). Since M lies on the line y = k, the y-coordinates of P and Q are both equal to k.
Therefore, the x-coordinate of M is given by:
(x1 + x2)/2 = x-coordinate of M
2. Midpoint N:
Since the line x = h is the perpendicular bisector of RQ, N lies on the line x = h. Therefore, the x-coordinate of N is equal to h.
Since N is the midpoint of RQ, we can find its y-coordinate by taking the average of the y-coordinates of R and Q. The coordinates of R are (-h, -k), so the y-coordinate of N is given by:
(-k + y2)/2 = y-coordinate of N
Finding the Coordinates of P:
Since M and N are midpoints, we can equate their coordinates to find the coordinates of P:
x-coordinate of M = x-coordinate of N
(y-coordinate of P + k)/2 = (-k + y2)/2
Simplifying the equation:
y-coordinate of P + k = -k + y2
Since the y-coordinates of P and Q are both equal to k, we have:
k + k = -k + y2
Simplifying further:
2k = -k + y2
Rearranging the equation:
3k = y2
Since the y-coordinate of P is 3k, the coordinates of P are (3h, 3k).
Therefore, the correct answer is option D: (3h, 3k).
The Perpendicular Bisector Property:
A line is the perpendicular bisector of a line segment if it passes through the midpoint of the segment and is perpendicular to it.
Given Information:
1. The line y = k is the perpendicular bisector of the line segment PQ.
2. The line x = h is the perpendicular bisector of the line segment RQ.
3. Point R has coordinates (-h, -k).
Using the Perpendicular Bisector Property:
Since the line y = k is the perpendicular bisector of line segment PQ, the midpoint of PQ lies on the line y = k. Let the midpoint be M.
Since the line x = h is the perpendicular bisector of line segment RQ, the midpoint of RQ lies on the line x = h. Let the midpoint be N.
Now, let's find the coordinates of M and N:
1. Midpoint M:
Since the line y = k is the perpendicular bisector of PQ, M lies on the line y = k. Therefore, the y-coordinate of M is equal to k.
Since M is the midpoint of PQ, we can find its x-coordinate by taking the average of the x-coordinates of P and Q. Let's assume the coordinates of P are (x1, y1) and the coordinates of Q are (x2, y2). Since M lies on the line y = k, the y-coordinates of P and Q are both equal to k.
Therefore, the x-coordinate of M is given by:
(x1 + x2)/2 = x-coordinate of M
2. Midpoint N:
Since the line x = h is the perpendicular bisector of RQ, N lies on the line x = h. Therefore, the x-coordinate of N is equal to h.
Since N is the midpoint of RQ, we can find its y-coordinate by taking the average of the y-coordinates of R and Q. The coordinates of R are (-h, -k), so the y-coordinate of N is given by:
(-k + y2)/2 = y-coordinate of N
Finding the Coordinates of P:
Since M and N are midpoints, we can equate their coordinates to find the coordinates of P:
x-coordinate of M = x-coordinate of N
(y-coordinate of P + k)/2 = (-k + y2)/2
Simplifying the equation:
y-coordinate of P + k = -k + y2
Since the y-coordinates of P and Q are both equal to k, we have:
k + k = -k + y2
Simplifying further:
2k = -k + y2
Rearranging the equation:
3k = y2
Since the y-coordinate of P is 3k, the coordinates of P are (3h, 3k).
Therefore, the correct answer is option D: (3h, 3k).
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In the xy-plane, the line y = k is the perpendicular bisector of the line segment PQ and the line x = h is the perpendicular bisector of the line segment RQ. If the coordinates of the point R are (-h, -k), then what are the coordinates of the point P?a)(-5h, -5k)b)(-3h, -3k)c)(2h, 2k)d)(3h, 3k)e)(5h, 5k)Correct answer is option 'D'. Can you explain this answer?
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In the xy-plane, the line y = k is the perpendicular bisector of the line segment PQ and the line x = h is the perpendicular bisector of the line segment RQ. If the coordinates of the point R are (-h, -k), then what are the coordinates of the point P?a)(-5h, -5k)b)(-3h, -3k)c)(2h, 2k)d)(3h, 3k)e)(5h, 5k)Correct answer is option 'D'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about In the xy-plane, the line y = k is the perpendicular bisector of the line segment PQ and the line x = h is the perpendicular bisector of the line segment RQ. If the coordinates of the point R are (-h, -k), then what are the coordinates of the point P?a)(-5h, -5k)b)(-3h, -3k)c)(2h, 2k)d)(3h, 3k)e)(5h, 5k)Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the xy-plane, the line y = k is the perpendicular bisector of the line segment PQ and the line x = h is the perpendicular bisector of the line segment RQ. If the coordinates of the point R are (-h, -k), then what are the coordinates of the point P?a)(-5h, -5k)b)(-3h, -3k)c)(2h, 2k)d)(3h, 3k)e)(5h, 5k)Correct answer is option 'D'. Can you explain this answer?.
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ample number of questions to practice In the xy-plane, the line y = k is the perpendicular bisector of the line segment PQ and the line x = h is the perpendicular bisector of the line segment RQ. If the coordinates of the point R are (-h, -k), then what are the coordinates of the point P?a)(-5h, -5k)b)(-3h, -3k)c)(2h, 2k)d)(3h, 3k)e)(5h, 5k)Correct answer is option 'D'. Can you explain this answer? tests, examples and also practice GMAT tests.
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