To determine if two lines are perpendicular, you can use the slopes of the lines. Specifically, two lines are perpendicular if the product of their slopes is -1. This is derived from the fact that the angle between two perpendicular lines is 90 degrees.

### Mathematical Explanation

Let's consider two lines with slopes \(m_1\) and \(m_2\).

1. **Slope Formula:** The slope of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]

2. **Condition for Perpendicularity:** For two lines to be perpendicular, the product of their slopes must be -1:
   \[
   m_1 \times m_2 = -1
   \]

### Example Calculation

Let's apply this to our previous example with quadrilateral WXYZ.

#### Coordinates:
- W \((1, 1)\)
- X \((4, 4)\)
- Y \((7, 2)\)
- Z \((4, -1)\)

#### Slopes Calculation:

1. **Slope of WX:**
   \[
   m_{WX} = \frac{4 - 1}{4 - 1} = \frac{3}{3} = 1
   \]

2. **Slope of XY:**
   \[
   m_{XY} = \frac{2 - 4}{7 - 4} = \frac{-2}{3}
   \]

3. **Slope of YZ:**
   \[
   m_{YZ} = \frac{-1 - 2}{4 - 7} = \frac{-3}{-3} = 1
   \]

4. **Slope of ZW:**
   \[
   m_{ZW} = \frac{1 - (-1)}{1 - 4} = \frac{2}{-3}
   \]

#### Check Perpendicularity:

1. **WX and XY:**
   \[
   m_{WX} \times m_{XY} = 1 \times \left(-\frac{2}{3}\right) = -\frac{2}{3}
   \]
   Since \(-\frac{2}{3} \neq -1\), WX and XY are not perpendicular.

2. **XY and YZ:**
   \[
   m_{XY} \times m_{YZ} = \left(-\frac{2}{3}\right) \times 1 = -\frac{2}{3}
   \]
   Since \(-\frac{2}{3} \neq -1\), XY and YZ are not perpendicular.

3. **YZ and ZW:**
   \[
   m_{YZ} \times m_{ZW} = 1 \times \left(-\frac{2}{3}\right) = -\frac{2}{3}
   \]
   Since \(-\frac{2}{3} \neq -1\), YZ and ZW are not perpendicular.

4. **ZW and WX:**
   \[
   m_{ZW} \times m_{WX} = \left(-\frac{2}{3}\right) \times 1 = -\frac{2}{3}
   \]
   Since \(-\frac{2}{3} \neq -1\), ZW and WX are not perpendicular.

Therefore, based on the slope calculations, none of the adjacent sides are perpendicular. This means that WXYZ is not a rectangle.

---

Do you have any other questions or need further clarification on this topic?

Here are some related questions:
1. What is the significance of the slope in determining the nature of a quadrilateral?
2. How can you find the midpoint of a line segment in a coordinate plane?
3. How do you calculate the length of a diagonal in a rectangle?
4. What are the properties of a parallelogram?
5. How can the distance formula be used to verify the sides of a quadrilateral?
6. What are the necessary conditions for a quadrilateral to be a square?
7. Can you use trigonometry to determine if angles in a quadrilateral are right angles?
8. How does the Pythagorean theorem apply to coordinate geometry?

**Tip:** When analyzing geometric figures on a coordinate plane, double-check all slope calculations and ensure that you accurately apply the conditions for parallelism and perpendicularity.

Explore how to determine if two lines are perpendicular using slope calculations. Learn about the mathematical condition for perpendicularity and apply it to a geometric example. Suitable for Grades 10-12 students studying coordinate geometry.

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