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The problem is about a rectangle on a coordinate plane. You are given two points, \( P(-8, 9) \) and \( Q(11, 9) \), which form one side of the rectangle. The perimeter of the rectangle is 52 units. You are asked to identify all possible points that could be vertices of the rectangle from the options provided.

### Step 1: Length of side PQ
- The segment from \( P(-8, 9) \) to \( Q(11, 9) \) is a horizontal line because the y-coordinates are the same (both 9). To find the length of this side:
  \[
  \text{Length of PQ} = |11 - (-8)| = |11 + 8| = 19 \text{ units}
  \]

### Step 2: Perimeter of rectangle
- The perimeter \( P \) of a rectangle is given by:
  \[
  P = 2 \times (\text{Length} + \text{Width})
  \]
  We know the perimeter is 52 units, and the length (side PQ) is 19 units. Let the unknown width be \( w \).
  \[
  52 = 2 \times (19 + w)
  \]
  Simplifying:
  \[
  52 = 38 + 2w
  \]
  \[
  2w = 14 \quad \Rightarrow \quad w = 7 \text{ units}
  \]

### Step 3: Identifying possible vertices
The rectangle must have a width of 7 units, meaning the remaining points must be 7 units away vertically from the line \( y = 9 \). So, the possible y-coordinates for the other two vertices are \( y = 9 + 7 = 16 \) and \( y = 9 - 7 = 2 \).

### Step 4: Verifying the given options
Let's check which points could be valid vertices by looking at their coordinates:

1. \( (16, 11) \): This point is not 7 units away from \( y = 9 \), so it cannot be a vertex.
2. \( (11, 4) \): The y-coordinate here is not 7 units away from \( y = 9 \), so it is not a valid vertex.
3. \( (-8, 2) \): The y-coordinate \( 2 \) is 7 units away from \( y = 9 \), and the x-coordinate matches the x-coordinate of \( P \), so this is a valid vertex.
4. \( (11, 16) \): The y-coordinate \( 16 \) is 7 units away from \( y = 9 \), and the x-coordinate matches the x-coordinate of \( Q \), so this is a valid vertex.

### Conclusion:
The valid vertices are:
- \( (-8, 2) \)
- \( (11, 16) \)

Would you like any further explanation or details?

Here are five related questions to help expand on this concept:
1. How do you calculate the distance between two points on a coordinate plane?
2. Can you explain how the perimeter formula for a rectangle applies to different shapes?
3. How would you calculate the area of this rectangle?
4. How can you find the diagonal length of the rectangle?
5. How would the problem change if the rectangle were rotated?

**Tip:** When solving geometry problems on the coordinate plane, always check if the sides are vertical or horizontal by examining the coordinates of points.

On the coordinate plane, the segment from P(-8, 9) to Q(11, 9) forms one side of a rectangle. The rectangle has a perimeter of 52 units. Select all of the points that could be a vertex of the rectangle.

This problem involves determining the possible vertices of a rectangle on a coordinate plane given two points and the perimeter. It includes the calculation of side lengths, distance formulas, and verification of valid vertex coordinates.

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