To prove that $AB$ is perpendicular to $CD$, we need to show that the slopes of these two line segments multiply to $-1$. This is because two lines are perpendicular if and only if the product of their slopes is $-1$.

Step 1: Calculate the Slope of $AB$

Given the points:

• $A(2, 1)$
• $B(-4, -3)$

The slope $m_{AB}$ is calculated using the formula:

$m_{AB} = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the coordinates of $A$ and $B$:

$m_{AB} = \frac{-3 - 1}{-4 - 2} = \frac{-4}{-6} = \frac{2}{3}$

Step 2: Calculate the Slope of $CD$

Given the points:

• $C(1, -4)$
• $D(-1, -1)$

The slope $m_{CD}$ is calculated using the formula:

$m_{CD} = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the coordinates of $C$ and $D$:

$m_{CD} = \frac{-1 - (-4)}{-1 - 1} = \frac{-1 + 4}{-2} = \frac{3}{-2} = -\frac{3}{2}$

Step 3: Verify if $AB$ is Perpendicular to $CD$

To check if the lines are perpendicular, multiply the slopes $m_{AB}$ and $m_{CD}$:

$m_{AB} \times m_{CD} = \frac{2}{3} \times -\frac{3}{2} = -\frac{6}{6} = -1$

Since the product of the slopes is $-1$, it follows that line segments $AB$ and $CD$ are perpendicular. Therefore, $AB \perp CD$ is proven.

Would you like more details or have any questions? Here are some related questions:

1. What are the coordinates of the point where lines $AB$ and $CD$ intersect?
2. How would you find the length of the perpendicular from point $C$ to line $AB$?
3. Can you determine the angle between $AB$ and $CD$?
4. How would the analysis change if one of the segments was vertical or horizontal?
5. What is the equation of the line perpendicular to $AB$ and passing through point $C$?

Tip: When proving perpendicularity, always ensure to check the slopes of the lines, as the product of the slopes will guide you to the correct conclusion.