4.3 Q-8

Question Statement

Two pairs of points are given. Find whether the two lines determined by these points are:
(i) parallel
(ii) perpendicular
(iii) neither.

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Background and Explanation

To determine whether two lines are parallel, perpendicular, or neither, we calculate their slopes.

1. Parallel Lines: Two lines are parallel if their slopes are equal.
2. Perpendicular Lines: Two lines are perpendicular if the product of their slopes is equal to -1.
   In both cases, we’ll use the formula for the slope of a line passing through two points:
   $m = \frac{y_2 - y_1}{x_2 - x_1}$
   Where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

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Solution

Part (a)

We are given points $A(1, -2), B(2, 4), C(4, 1)$. We need to determine if the lines $\overline{AB}$ and $\overline{CD}$ are parallel, perpendicular, or neither.

Step 1: Find the slope of $\overline{AB}$

Using the formula for slope:
$m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-2)}{2 - 1} = \frac{6}{1} = 6$
So, the slope of $\overline{AB}$ is $m_1 = 6$.

Step 2: Find the slope of $\overline{CD}$

Using the same formula:
$m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-2)}{4 - 1} = \frac{3}{3} = 1$
So, the slope of $\overline{CD}$ is $m_2 = 1$.

Step 3: Compare the slopes

We can see that:
$m_1 = 6 \quad \text{and} \quad m_2 = 1$
Since the slopes are not equal, the lines are not parallel.

To check if the lines are perpendicular, we multiply the slopes:
$m_1 \times m_2 = 6 \times 1 = 6$
Since the product is not equal to $-1$, the lines are not perpendicular.

Thus, the lines are neither parallel nor perpendicular.

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Part (b)

We are given points $A(-3, 4), B(6, 2), C(4, 5), D(-2, -7)$. We need to determine if the lines $\overline{AB}$ and $\overline{CD}$ are parallel, perpendicular, or neither.

Step 1: Find the slope of $\overline{AB}$

Using the slope formula:
$m_1 = \frac{2 - 4}{6 - (-3)} = \frac{-2}{9}$
So, the slope of $\overline{AB}$ is $m_1 = \frac{-2}{9}$.

Step 2: Find the slope of $\overline{CD}$

Using the formula for the slope:
$m_2 = \frac{-7 - 5}{-2 - 4} = \frac{-12}{-6} = 2$
So, the slope of $\overline{CD}$ is $m_2 = 2$.

Step 3: Compare the slopes

We can see that:
$m_1 = \frac{-2}{9} \quad \text{and} \quad m_2 = 2$
Since the slopes are not equal, the lines are not parallel.

To check if the lines are perpendicular, we multiply the slopes:
$m_1 \times m_2 = \frac{-2}{9} \times 2 = \frac{-4}{9}$
Since the product is not equal to $-1$, the lines are not perpendicular.

Thus, the lines are neither parallel nor perpendicular.

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Key Formulas or Methods Used

• Slope formula:
  $m = \frac{y_2 - y_1}{x_2 - x_1}$
• Parallel Lines Condition: Lines are parallel if $m_1 = m_2$.
• Perpendicular Lines Condition: Lines are perpendicular if $m_1 \times m_2 = -1$.

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Summary of Steps

1. Calculate the slopes of both lines using the slope formula.
2. Check if the slopes are equal (parallel lines) or if their product is -1 (perpendicular lines).
3. Conclude whether the lines are parallel, perpendicular, or neither based on the comparison of slopes.