Quadrilateral PQRS has coordinates P(8, -8), Q(2, -8), and R(2, -4). Which ordered pair for point S would give the quadrilateral the greatest perimeter

To find the ordered pair for point \( S \) that would give the quadrilateral \( PQRS \) the greatest perimeter, let's explore the possible scenarios. 

### Given Points:
- \( P(8, -8) \)
- \( Q(2, -8) \)
- \( R(2, -4) \)

### Distance Formulas:
First, let's calculate the lengths of the sides \( PQ \), \( QR \), and \( RP \).

1. **Length of \( PQ \):**
   \[
   PQ = \sqrt{(2 - 8)^2 + (-8 - (-8))^2} = \sqrt{(-6)^2 + 0^2} = \sqrt{36} = 6
   \]

2. **Length of \( QR \):**
   \[
   QR = \sqrt{(2 - 2)^2 + (-4 - (-8))^2} = \sqrt{0^2 + 4^2} = \sqrt{16} = 4
   \]

3. **Length of \( RP \):**
   \[
   RP = \sqrt{(8 - 2)^2 + (-8 - (-4))^2} = \sqrt{6^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}
   \]

### Maximize Perimeter:
To maximize the perimeter, we want the length \( SR \) to be as large as possible while still being connected to point \( R(2, -4) \). Since the quadrilateral has 4 sides, we calculate possible values for \( S(x, y) \) and their effect on the perimeter.

#### Potential Locations for \( S \):
1. **\( S_1(8, -4) \)**: Placing \( S \) directly above \( R \) on the same vertical line as \( P \).
   - \( PS_1 = 4 \)
   - \( S_1R = \sqrt{(8 - 2)^2 + (-4 - (-4))^2} = \sqrt{36} = 6 \)
   - Perimeter = \( PQ + QR + RP + PS_1 + S_1R \)
   - Perimeter = \( 6 + 4 + 2\sqrt{13} + 4 + 6 \)

2. **\( S_2(8, -12) \)**: Placing \( S \) in a way that aligns vertically downward from \( P \), but on the same line horizontally with \( R \).
   - \( PS_2 = \sqrt{(8 - 8)^2 + (-8 - (-12))^2} = 4 \)
   - \( S_2R = \sqrt{(8 - 2)^2 + (-12 - (-4))^2} = \sqrt{36 + 64} = 10 \)
   - Perimeter = \( 6 + 4 + 2\sqrt{13} + 4 + 10 \)

3. **\( S_3(2, -12) \)**: Placing \( S \) directly downward from \( R \) on the same vertical line.
   - \( PS_3 = \sqrt{(8 - 2)^2 + (-8 - (-12))^2} = \sqrt{36 + 16} = 2\sqrt{13} \)
   - \( S_3R = 8 \) (as \( S_3 \) and \( R \) have the same \( x \) value).
   - Perimeter = \( 6 + 4 + 2\sqrt{13} + 2\sqrt{13} + 8 \)

### Conclusion:
The scenario that maximizes the perimeter is when \( S(2, -12) \) as \( S_3(2, -12) \) gives the greatest side \( SR \), leading to the greatest overall perimeter.

Thus, **the ordered pair for point \( S \) that would give the quadrilateral the greatest perimeter is \( S(2, -12) \)**.

Would you like to explore any other details or have any questions? 

Here are 5 related questions:
1. How would the perimeter change if \( S \) were placed at \( (4, -10) \)?
2. What if the shape were a rectangle instead of a general quadrilateral?
3. How do you determine the area of quadrilateral \( PQRS \) with any chosen \( S \)?
4. How would the perimeter compare if we positioned \( S \) at \( (10, -10) \)?
5. What impact does the distance from \( R \) to \( S \) have on maximizing perimeter?

**Tip:** When maximizing perimeter, always consider the farthest possible point relative to the base of the shape.

Explore how to maximize the perimeter of quadrilateral PQRS with coordinates P(8, -8), Q(2, -8), and R(2, -4) by determining the optimal location for point S. Learn about coordinate geometry, distance formulas, and perimeter calculations. Suitable for students in Grades 7-9.

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