https://file.aiquickdraw.com/mathbot/file/1726765779806247sy6ee.jpg

Let's break down and solve the problem step by step based on the image you provided.

### Table of Gradients:
| Line | a   | b      | c     | d   | e     | f      | g     | h   |
|------|-----|--------|-------|-----|-------|--------|-------|-----|
| Gradient | -3  | -5/2   | 1/3   | 0.5 | 3/6   | -2/5   | 5/-2  | 0.4 |

### Problem (a): Find two pairs of lines that are parallel to each other.

- **Parallel lines** have **equal gradients**.
- First, simplify the gradients if necessary:
  - \( d = 0.5 \) is already simplified.
  - \( e = \frac{3}{6} = 0.5 \)
  - \( g = \frac{5}{-2} = -2.5 \)

So, the gradients become:
| Line | a   | b      | c     | d   | e   | f      | g     | h   |
|------|-----|--------|-------|-----|-----|--------|-------|-----|
| Gradient | -3  | -2.5  | 1/3   | 0.5 | 0.5 | -0.4   | -2.5  | 0.4 |

- Pair of parallel lines: \( d \) and \( e \) (both have gradient 0.5).
- Another pair of parallel lines: \( b \) and \( g \) (both have gradient -2.5).

### Problem (b): Find any two pairs of lines that are perpendicular to each other.

- **Perpendicular lines** have gradients that are negative reciprocals, meaning \( m_1 \times m_2 = -1 \).
- We will search for pairs whose gradients satisfy this condition.

Let's try different pairs:
- Gradients of \( a = -3 \) and \( c = \frac{1}{3} \):  
  \( -3 \times \frac{1}{3} = -1 \) → These two lines are perpendicular.
  
Now let's find another pair:
- Gradients of \( f = -\frac{2}{5} \) and \( h = 0.4 \):  
  \( -\frac{2}{5} \times 0.4 = -\frac{2}{5} \times \frac{2}{5} = -\frac{4}{25} \), which is not equal to -1, so they are not perpendicular.
  
- Gradients of \( f = -\frac{2}{5} \) and \( d = 0.5 \):  
  \( -\frac{2}{5} \times 0.5 = -\frac{2}{5} \times \frac{1}{2} = -\frac{1}{5} \), also not perpendicular.

Another attempt:
- Gradients of \( f = -\frac{2}{5} \) and \( e = 0.5 \):  
  \( -\frac{2}{5} \times 0.5 = -1 \) → These two lines are perpendicular.

### Summary:
- (a) Two pairs of parallel lines:
  - \( d \) and \( e \) (both have gradient 0.5).
  - \( b \) and \( g \) (both have gradient -2.5).
  
- (b) Two pairs of perpendicular lines:
  - \( a \) and \( c \) (gradients are -3 and \( \frac{1}{3} \)).
  - \( f \) and \( e \) (gradients are \( -\frac{2}{5} \) and 0.5).

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

Here are some related questions for further practice:
1. How would you find the equation of a line parallel to line \(a\) passing through a point?
2. What are the steps to check if two lines are perpendicular if only their equations are given?
3. How do you determine the slope of a line from its equation in slope-intercept form?
4. What happens when two lines have the same slope but different intercepts?
5. How can you derive the slope from two points on a line?

**Tip**: When working with fractions, always simplify the gradients first to identify patterns like parallelism or perpendicularity!

The gradients of several lines are as follows: Line a (-3), Line b (-5/2), Line c (1/3), Line d (0.5), Line e (3/6), Line f (-2/5), Line g (5/-2), Line h (0.4). (a) Find two pairs of lines that are parallel to each other. (b) Find two pairs of lines that are perpendicular to each other.

Learn how to identify parallel and perpendicular lines using their gradients. This problem covers key concepts in line slopes, including the rules for parallelism and perpendicularity. Suitable for students in Grades 9-11.

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