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In the xy-plane, the points \(\mathrm{(c, d)}\), \(\mathrm{(c, -d)}\), and \(\mathrm{(-c, -d)}\) are three vertices of a certain square. If \(\mathrm{c < 0}\) and \(\mathrm{d > 0}\), which of the following points is in the same quadrant as the fourth vertex of the square?
Let's understand what we're working with. We have three points that are vertices of a square: \((\mathrm{c}, \mathrm{d})\), \((\mathrm{c}, -\mathrm{d})\), and \((-\mathrm{c}, -\mathrm{d})\). We also know that \(\mathrm{c} < 0\) (meaning c is negative) and \(\mathrm{d} > 0\) (meaning d is positive).
Since this is about finding vertices of a square, we need to remember that a square has four vertices, and we're given three of them. Our job is to find where the fourth vertex is located and determine which quadrant it's in.
Process Skill: TRANSLATE - Converting the coordinate conditions into a clear geometric picture
Let's place these three points on the coordinate plane. Remember that c is negative and d is positive.
Point 1: \((\mathrm{c}, \mathrm{d})\) - Since \(\mathrm{c} < 0\) and \(\mathrm{d} > 0\), this point is in Quadrant II (negative x, positive y)
Point 2: \((\mathrm{c}, -\mathrm{d})\) - Since \(\mathrm{c} < 0\) and \(-\mathrm{d} < 0\), this point is in Quadrant III (negative x, negative y)
Point 3: \((-\mathrm{c}, -\mathrm{d})\) - Since \(-\mathrm{c} > 0\) and \(-\mathrm{d} < 0\), this point is in Quadrant IV (positive x, negative y)
Notice something interesting: we have points in three different quadrants (II, III, and IV), but none in Quadrant I yet.
Process Skill: VISUALIZE - Creating a mental picture of where these points are located
Now let's think about how squares work. If we look at our three points, we can see a pattern:
This suggests that \((\mathrm{c}, \mathrm{d})\) and \((-\mathrm{c}, -\mathrm{d})\) are diagonally opposite each other in the square. Similarly, \((\mathrm{c}, -\mathrm{d})\) should be diagonally opposite to our missing fourth vertex.
If \((\mathrm{c}, -\mathrm{d})\) is at coordinates \((\mathrm{c}, -\mathrm{d})\), then its diagonal opposite should be at \((-\mathrm{c}, \mathrm{d})\).
Let's verify this makes sense: if our fourth vertex is \((-\mathrm{c}, \mathrm{d})\), then we have:
This confirms our fourth vertex is at \((-\mathrm{c}, \mathrm{d})\).
Our fourth vertex is at \((-\mathrm{c}, \mathrm{d})\). Let's determine its quadrant:
So our fourth vertex \((-\mathrm{c}, \mathrm{d})\) has coordinates (positive, positive), which places it in Quadrant I.
Looking at our answer choices, we need to find which one is also in Quadrant I (positive x, positive y):
The fourth vertex is located at \((-\mathrm{c}, \mathrm{d})\), which is in Quadrant I since \(-\mathrm{c} > 0\) and \(\mathrm{d} > 0\). Among the answer choices, only option E \((3, 5)\) is also in Quadrant I.
The answer is E.
1. Misinterpreting the constraint conditions: Students may confuse the signs of c and d. Since \(\mathrm{c} < 0\) and \(\mathrm{d} > 0\), students might incorrectly place the given points in the wrong quadrants by mixing up which variable is positive or negative.
2. Assuming incorrect square orientation: Students may assume the square is axis-aligned (sides parallel to x and y axes) without carefully analyzing the given vertices. They might try to form a square by simply adding/subtracting the same values to coordinates rather than understanding the geometric relationship.
3. Misunderstanding the square's vertex relationship: Students may not recognize that in a square, vertices can be identified by their diagonal relationships or may incorrectly assume which vertices are adjacent versus diagonal to each other.
1. Sign errors when finding the fourth vertex: When determining that the fourth vertex is \((-\mathrm{c}, \mathrm{d})\), students may make sign errors, particularly forgetting that if \(\mathrm{c} < 0\), then \(-\mathrm{c} > 0\). This could lead them to place the fourth vertex in the wrong quadrant.
2. Incorrect quadrant identification: Students may correctly find the fourth vertex as \((-\mathrm{c}, \mathrm{d})\) but then incorrectly determine which quadrant it belongs to, especially when dealing with the negative of a negative number (-c when \(\mathrm{c} < 0\)).
1. Quadrant classification errors for answer choices: Even after correctly determining that the fourth vertex is in Quadrant I, students may incorrectly classify the quadrants of the given answer choices, particularly confusing the signs of coordinates in options like \((3, -5)\) versus \((3, 5)\).