Line 

ℓ

1

 has the equation 

y

=

1

2

x

+

9

2

 and line 

ℓ

2

 has the equation 

–

x

+

2

y

=

–

4

.

 Find the distance between 

ℓ

1

and 

ℓ

2

.

Round your answer to the nearest tenth.




To find the distance between the two lines, we need to ensure both lines are in the slope-intercept form \( y = mx + c \). 

The first line \( \ell_1 \) is already in this form:

\[
\ell_1: y = \frac{1}{2}x + \frac{9}{2}
\]

The second line \( \ell_2 \) is given by:

\[
-x + 2y = -4
\]

Let's rearrange \( \ell_2 \) into the slope-intercept form \( y = mx + c \). Start by solving for \( y \):

\[
2y = x - 4
\]

\[
y = \frac{1}{2}x - 2
\]

Now, we have:

\[
\ell_2: y = \frac{1}{2}x - 2
\]

Both lines \( \ell_1 \) and \( \ell_2 \) have the same slope \( m = \frac{1}{2} \). We can use the formula for the distance \( d \) between two parallel lines \( y = mx + c_1 \) and \( y = mx + c_2 \):

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

Here, \( m = \frac{1}{2} \), \( c_1 = \frac{9}{2} \), and \( c_2 = -2 \). Substituting these values into the formula:

\[
d = \frac{\left|\frac{9}{2} - (-2)\right|}{\sqrt{1 + \left(\frac{1}{2}\right)^2}}
\]

First, simplify the expression inside the absolute value and calculate the denominator:

\[
d = \frac{\left|\frac{9}{2} + \frac{4}{2}\right|}{\sqrt{1 + \frac{1}{4}}} = \frac{\left|\frac{13}{2}\right|}{\sqrt{\frac{5}{4}}}
\]

Simplify further:

\[
d = \frac{\frac{13}{2}}{\frac{\sqrt{5}}{2}} = \frac{13}{\sqrt{5}}
\]

Now, compute \( \frac{13}{\sqrt{5}} \):

\[
d = \frac{13}{\sqrt{5}} \approx \frac{13}{2.236} \approx 5.813
\]

Rounding to the nearest tenth:

\[
d \approx 5.8
\]

So, the distance between the lines \( \ell_1 \) and \( \ell_2 \) is approximately **5.8 units**.

Would you like more details or have any questions?

Here are some related questions:

1. How do you convert the standard form of a line equation to the slope-intercept form?
2. What happens if the lines are perpendicular instead of parallel?
3. Can you calculate the distance between two skew lines in three dimensions?
4. How would the distance formula change if one line was horizontal or vertical?
5. How does the y-intercept affect the distance between two parallel lines?

**Tip:** Always verify that both lines are parallel before using the distance formula between parallel lines. If they aren’t parallel, the distance will be undefined by this method.

Learn how to find the distance between two lines given their equations: y = (1/2)x + 9/2 and -x + 2y = -4. Follow a step-by-step solution using the distance formula for parallel lines. Suitable for high school students studying geometry.

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