To determine which lines are perpendicular to each other, we can compare the slopes of the given lines. Two lines are perpendicular if the product of their slopes is $-1$.

Step 1: Write the equations of the lines in slope-intercept form ($y = mx + b$), where $m$ is the slope.

(a) $-3x + 4y + 4 = 0$

Solve for $y$: $4y = 3x - 4$ $y = \frac{3}{4}x - 1$ Slope of (a) = $m_1 = \frac{3}{4}$.

(b) $-4x - 3y = 0$

Solve for $y$: $-3y = 4x$ $y = -\frac{4}{3}x$ Slope of (b) = $m_2 = -\frac{4}{3}$.

(c) $-3x - 4y = 3$

Solve for $y$: $-4y = 3x - 3$ $y = -\frac{3}{4}x + \frac{3}{4}$ Slope of (c) = $m_3 = -\frac{3}{4}$.

(d) $4x + 3y + 5 = 0$

Solve for $y$: $3y = -4x - 5$ $y = -\frac{4}{3}x - \frac{5}{3}$ Slope of (d) = $m_4 = -\frac{4}{3}$.

(e) $3x - 4y + 9 = 0$

Solve for $y$: $-4y = -3x - 9$ $y = \frac{3}{4}x + \frac{9}{4}$ Slope of (e) = $m_5 = \frac{3}{4}$.

(f) $-4x + 3y - 4 = 0$

Solve for $y$: $3y = 4x + 4$ $y = \frac{4}{3}x + \frac{4}{3}$ Slope of (f) = $m_6 = \frac{4}{3}$.

Step 2: Check perpendicularity

Two lines are perpendicular if the product of their slopes is $-1$. Let’s check for each pair:

• Slopes of (a) and (b): $\frac{3}{4} \times -\frac{4}{3} = -1$ (Perpendicular)
• Slopes of (a) and (d): $\frac{3}{4} \times -\frac{4}{3} = -1$ (Perpendicular)
• Slopes of (c) and (f): $-\frac{3}{4} \times \frac{4}{3} = -1$ (Perpendicular)

Conclusion:

• Line (a) is perpendicular to line (b).
• Line (a) is perpendicular to line (d).
• Line (c) is perpendicular to line (f).

Would you like more details on the calculations, or do you have any other questions?

Here are 5 related questions you might find interesting:

1. How can we find the equation of a line perpendicular to a given line and passing through a specific point?
2. What is the geometrical significance of two lines being perpendicular?
3. How do you find the slope of a line if the equation is given in general form?
4. Can parallel lines also be perpendicular to other lines?
5. How do you determine if two lines are parallel based on their slopes?

Tip: The product of the slopes of perpendicular lines is always $-1$, which is a quick way to check for perpendicularity!