Answer
Quadrilaterals PQSY and SRTZ are parallelograms. Quadrilateral WRYN cannot be determined as a parallelogram because the coordinates for point N are incomplete.
Solution
To determine whether each given quadrilateral is a parallelogram, we'll analyze the coordinates of their vertices. A quadrilateral is a **parallelogram** if both pairs of opposite sides are either parallel (having the same slope) and equal in length or if the diagonals bisect each other (have the same midpoint).
Let's examine each quadrilateral one by one.
---
### **5. Quadrilateral PQSY**
**Vertices:**
- \( P(0, 0) \)
- \( Q(3, 4) \)
- \( S(7, 4) \)
- \( Y(4, 0) \)
**Steps to Determine if PQSY is a Parallelogram:**
1. **Calculate the Slopes of Opposite Sides:**
- **Slope of PQ:**
\[
\text{Slope}_{PQ} = \frac{4 - 0}{3 - 0} = \frac{4}{3}
\]
- **Slope of SY:**
\[
\text{Slope}_{SY} = \frac{0 - 4}{4 - 7} = \frac{-4}{-3} = \frac{4}{3}
\]
- **Slope of QS:**
\[
\text{Slope}_{QS} = \frac{4 - 4}{7 - 3} = 0
\]
- **Slope of PY:**
\[
\text{Slope}_{PY} = \frac{0 - 0}{4 - 0} = 0
\]
2. **Check for Parallelism:**
- **PQ is parallel to SY** because both have a slope of \( \frac{4}{3} \).
- **QS is parallel to PY** because both have a slope of \( 0 \).
3. **Verify Equal Lengths (Optional but Confirms Parallelogram):**
- **Length of PQ:**
\[
\sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = 5
\]
- **Length of SY:**
\[
\sqrt{(7-4)^2 + (4-0)^2} = \sqrt{9 + 16} = 5
\]
- **Length of QS:**
\[
\sqrt{(7-3)^2 + (4-4)^2} = \sqrt{16} = 4
\]
- **Length of PY:**
\[
\sqrt{(4-0)^2 + (0-0)^2} = \sqrt{16} = 4
\]
Since both pairs of opposite sides are parallel and equal in length, **PQSY is a parallelogram**.
---
### **6. Quadrilateral SRTZ**
**Vertices:**
- \( S(-2, 1) \)
- \( R(1, 3) \)
- \( T(2, 0) \)
- \( Z(-1, -2) \)
**Steps to Determine if SRTZ is a Parallelogram:**
1. **Calculate the Slopes of Opposite Sides:**
- **Slope of SR:**
\[
\text{Slope}_{SR} = \frac{3 - 1}{1 - (-2)} = \frac{2}{3}
\]
- **Slope of TZ:**
\[
\text{Slope}_{TZ} = \frac{-2 - 0}{-1 - 2} = \frac{-2}{-3} = \frac{2}{3}
\]
- **Slope of RT:**
\[
\text{Slope}_{RT} = \frac{0 - 3}{2 - 1} = \frac{-3}{1} = -3
\]
- **Slope of SZ:**
\[
\text{Slope}_{SZ} = \frac{1 - (-2)}{-2 - (-1)} = \frac{3}{-1} = -3
\]
2. **Check for Parallelism:**
- **SR is parallel to TZ** because both have a slope of \( \frac{2}{3} \).
- **RT is parallel to SZ** because both have a slope of \( -3 \).
3. **Verify Equal Lengths (Optional but Confirms Parallelogram):**
- **Length of SR:**
\[
\sqrt{(1 - (-2))^2 + (3 - 1)^2} = \sqrt{9 + 4} = \sqrt{13}
\]
- **Length of TZ:**
\[
\sqrt{(-1 - 2)^2 + (-2 - 0)^2} = \sqrt{9 + 4} = \sqrt{13}
\]
- **Length of RT:**
\[
\sqrt{(2 - 1)^2 + (0 - 3)^2} = \sqrt{1 + 9} = \sqrt{10}
\]
- **Length of SZ:**
\[
\sqrt{(-2 - (-1))^2 + (1 - (-2))^2} = \sqrt{1 + 9} = \sqrt{10}
\]
Since both pairs of opposite sides are parallel and equal in length, **SRTZ is a parallelogram**.
---
### **7. Quadrilateral WRYN**
**Vertices:**
- \( W(2, 5) \)
- \( R(3, 3) \)
- \( Y(-2, -3) \)
- \( N(-) \) *(Incomplete Coordinate)*
**Analysis:**
The coordinate for point \( N \) is incomplete in the provided data (\( N(-) \)). To determine whether quadrilateral **WRYN** is a parallelogram, we need the complete coordinates of all four vertices. Without the full set of coordinates, we cannot perform the necessary calculations to verify the properties required for a parallelogram.
---
### **Conclusion**
- **Quadrilateral PQSY (Problem 5):** **Parallelogram**
- **Quadrilateral SRTZ (Problem 6):** **Parallelogram**
- **Quadrilateral WRYN (Problem 7):** **Cannot Determine** due to incomplete coordinate information for point \( N \).
If you provide the complete coordinates for point \( N \), I can assist you further in determining whether **WRYN** is a parallelogram.
Reviewed and approved by the UpStudy tutoring team
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