https://file.aiquickdraw.com/mathbot/file/1728832389017yvdo2217.jpg

The problem in the image asks for the perpendicular distance from a point to a line, using the equation of the line and the point given in the problem.

Let's break it down:

### Given:
- The equation of the line is \( 3x + 4y - 1 = 0 \).
- The point from which the perpendicular distance to this line is required has the coordinates \( (2, 3) \).
- The length of the perpendicular is 2 units.

### Formula for the perpendicular distance:
The formula to calculate the perpendicular distance \( d \) from a point \( (x_1, y_1) \) to a line \( Ax + By + C = 0 \) is:

\[
d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
\]

### Substituting values:
From the equation \( 3x + 4y - 1 = 0 \), we have:
- \( A = 3 \),
- \( B = 4 \),
- \( C = -1 \),
and the point \( (x_1, y_1) = (2, 3) \).

Using the formula, we can substitute these values:

\[
d = \frac{|3(2) + 4(3) - 1|}{\sqrt{3^2 + 4^2}} = \frac{|6 + 12 - 1|}{\sqrt{9 + 16}} = \frac{|17|}{5} = \frac{17}{5}
\]

However, we know that the actual distance is 2 units, so we set up the equation:

\[
\frac{17}{5} = 2
\]

This is incorrect for the point provided. Thus, there's an inconsistency that might involve interpreting another condition or considering a different calculation.

If this analysis needs revision or more information, I can help clarify.

Do you want to go into the details or clarify any point? Here are 5 related questions for further learning:

1. How can we generalize the distance formula for lines in three dimensions?
2. How do perpendicular slopes relate to each other in geometry?
3. What role do absolute values play in the distance formula?
4. Why is the denominator in the distance formula \( \sqrt{A^2 + B^2} \)?
5. How does the distance formula change for vertical or horizontal lines?

**Tip**: When working with distances and lines, always double-check the coefficients of the equation of the line and the coordinates of the point to ensure correct calculations.

Find the perpendicular distance from the point (2,3) to the line 3x + 4y - 1 = 0. Given the distance options, select the correct answer.

Learn how to calculate the perpendicular distance from a point to a line using the distance formula. The problem involves the line 3x + 4y - 1 = 0 and the point (2, 3). Suitable for students in Grades 9-12.

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