In the xy-plane, the points \(\mathrm{(c, d)}\), \(\mathrm{(c, -d)}\), and \(\mathrm{(-c, -d)}\) are three vertices of a certain square. If...

1. Translate the problem requirements: We have three vertices of a square: \(\mathrm{c}, \mathrm{d})\), \((\mathrm{c}, -\mathrm{d})\), and \((-\mathrm{c}, -\mathrm{d})\) where \(\mathrm{c} < 0\) and \(\mathrm{d} > 0\). We need to find which quadrant contains the fourth vertex.
2. Visualize the given vertices and their quadrant locations: Since \(\mathrm{c} < 0\) and \(\mathrm{d} > 0\), determine which quadrants contain the three given points.
3. Identify the square's structure and find the missing vertex: Use the property that opposite vertices of a square are diagonally across from each other to locate the fourth vertex.
4. Determine the quadrant of the fourth vertex: Analyze the coordinates of the fourth vertex to identify its quadrant and match with the answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

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

2. Visualize the given vertices and their quadrant locations

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

3. Identify the square's structure and find the missing vertex

Now let's think about how squares work. If we look at our three points, we can see a pattern:

• \((\mathrm{c}, \mathrm{d})\) and \((\mathrm{c}, -\mathrm{d})\) have the same x-coordinate but opposite y-coordinates
• \((\mathrm{c}, -\mathrm{d})\) and \((-\mathrm{c}, -\mathrm{d})\) have the same y-coordinate but opposite x-coordinates

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:

• \((\mathrm{c}, \mathrm{d})\) and \((-\mathrm{c}, \mathrm{d})\): same y-coordinate, opposite x-coordinates ✓
• \((-\mathrm{c}, \mathrm{d})\) and \((-\mathrm{c}, -\mathrm{d})\): same x-coordinate, opposite y-coordinates ✓

This confirms our fourth vertex is at \((-\mathrm{c}, \mathrm{d})\).

4. Determine the quadrant of the fourth vertex

Our fourth vertex is at \((-\mathrm{c}, \mathrm{d})\). Let's determine its quadrant:

• Since \(\mathrm{c} < 0\), we have \(-\mathrm{c} > 0\) (positive)
• Since \(\mathrm{d} > 0\) (given), d remains positive

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):

1. \((-5, -3)\): negative x, negative y → Quadrant III
2. \((-5, 3)\): negative x, positive y → Quadrant II
3. \((5, -3)\): positive x, negative y → Quadrant IV
4. \((3, -5)\): positive x, negative y → Quadrant IV
5. \((3, 5)\): positive x, positive y → Quadrant I ✓

4. Final Answer

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.

Common Faltering Points

Errors while devising the approach

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.

Errors while executing the approach

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 \(\mathrm{c} < 0\)).

Errors while selecting the answer

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 \((3, 5)\).