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Find the distance between the two red parallel lines shown.

To find the distance between two parallel lines, we first need their equations. From the graph, the red lines appear to have the same slope, confirming that they are parallel.

To find the distance, we use the formula for the distance between two parallel lines:

\[
d = \frac{|c_2 - c_1|}{\sqrt{1 + m^2}}
\]

where:
- \( m \) is the slope of the lines,
- \( c_1 \) and \( c_2 \) are the y-intercepts of the two lines.

### Step 1: Determine the slope of the lines.
From the graph, we can observe the slope. Both lines seem to have a negative slope since they are declining from left to right.

By visual inspection, the lines go through points that can help us calculate the slope. For example, one line passes through the points \( (0, 5) \) and \( (1, 3) \). Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

\[
m = \frac{3 - 5}{1 - 0} = \frac{-2}{1} = -2
\]

So, the slope \( m = -2 \).

### Step 2: Identify the y-intercepts.
Looking at the y-axis, we can see that:
- One line crosses the y-axis at \( y = 5 \),
- The other line crosses the y-axis at \( y = 4 \).

Thus, the equations of the lines are:
1. \( y = -2x + 5 \)
2. \( y = -2x + 4 \)

### Step 3: Apply the distance formula.
Using the distance formula for parallel lines:

\[
d = \frac{|4 - 5|}{\sqrt{1 + (-2)^2}} = \frac{| -1 |}{\sqrt{1 + 4}} = \frac{1}{\sqrt{5}} = \frac{1}{2.236} \approx 0.447
\]

So, the distance between the two lines is approximately \( 0.447 \) units.

Would you like further clarification or details?

### Related Questions:
1. How do we find the distance between non-parallel lines?
2. What is the formula for the shortest distance from a point to a line?
3. How can we calculate the distance between skew lines in 3D space?
4. What is the significance of the slope in determining if two lines are parallel?
5. How would the distance formula change for vertical or horizontal parallel lines?

**Tip:** Always check the slope of two lines when determining if they are parallel. Parallel lines always have identical slopes.

Find the distance between the two red parallel lines shown.

Learn how to calculate the distance between two parallel lines using their slopes and y-intercepts. This example involves applying the distance formula for parallel lines, suitable for students in Grades 9-12.

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