If it was given that 𝐷(− 1; − 1), prove that 𝐴𝐵 ⊥ 𝐶𝐷.

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.

Learn how to prove that lines AB and CD are perpendicular using coordinate geometry. This problem involves calculating slopes and verifying perpendicularity, suitable for high school students.

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