What is the length of the line with a slope of 3/2 from the point (4, 6) to the y-axis?

To find the length of the line with a slope of \( \frac{3}{2} \) from the point \( (4, 6) \) to the y-axis, we can follow these steps:

### 1. Find the equation of the line
The general equation of a line in slope-intercept form is:

\[
y = mx + b
\]

where \( m \) is the slope and \( b \) is the y-intercept. Given the slope \( m = \frac{3}{2} \) and the point \( (4, 6) \), we can substitute these into the equation to find \( b \):

\[
6 = \frac{3}{2}(4) + b
\]

\[
6 = 6 + b
\]

\[
b = 0
\]

So, the equation of the line is:

\[
y = \frac{3}{2}x
\]

### 2. Determine where the line intersects the y-axis
The line intersects the y-axis where \( x = 0 \). Substitute \( x = 0 \) into the equation:

\[
y = \frac{3}{2}(0) = 0
\]

So, the line intersects the y-axis at the point \( (0, 0) \).

### 3. Calculate the length of the line segment
Now, calculate the distance between the points \( (4, 6) \) and \( (0, 0) \) using the distance formula:

\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the coordinates \( (x_1, y_1) = (4, 6) \) and \( (x_2, y_2) = (0, 0) \):

\[
\text{Distance} = \sqrt{(0 - 4)^2 + (0 - 6)^2}
\]

\[
\text{Distance} = \sqrt{(-4)^2 + (-6)^2}
\]

\[
\text{Distance} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}
\]

Thus, the length of the line segment is \( 2\sqrt{13} \).

Would you like more details on any of the steps, or do you have any further questions?

Here are 5 related questions to explore further:

1. How do you find the equation of a line passing through two given points?
2. What is the significance of the y-intercept in the equation of a line?
3. How can you determine if two lines are parallel or perpendicular?
4. How do you find the midpoint of a line segment between two points?
5. What is the formula for the slope of a line given two points?

**Tip:** The slope of a line indicates the steepness and direction of the line. A positive slope means the line rises as it moves from left to right.

Learn how to calculate the length of a line segment with a slope of 3/2 from point (4, 6) to the y-axis using the distance formula. This problem involves coordinate geometry and the application of the distance formula. Suitable for high school students.

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