https://file.aiquickdraw.com/mathbot/file/17230465068568lnci68m.jpg

To find the distance from the point \((2, 1)\) to the line \(4x - 3y + 5 = 0\), we use the formula for the distance from a point to a line in the plane. The formula is given by:

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

where \((x_1, y_1)\) is the point and \(Ax + By + C = 0\) is the line.

For our specific problem:
- The point is \((2, 1)\).
- The line equation is \(4x - 3y + 5 = 0\), so \(A = 4\), \(B = -3\), and \(C = 5\).

Substitute these values into the formula:

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

Calculate the numerator:

\[
|4(2) - 3(1) + 5| = |8 - 3 + 5| = |10| = 10
\]

Calculate the denominator:

\[
\sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]

Thus, the distance \(d\) is:

\[
d = \frac{10}{5} = 2
\]

Therefore, the distance from the point \((2, 1)\) to the line \(4x - 3y + 5 = 0\) is \(2\).

Would you like any more details or have any questions?

Here are some related questions you might find interesting:
1. How do you derive the formula for the distance from a point to a line?
2. What is the geometric interpretation of the distance from a point to a line?
3. How would the distance change if the line equation was different?
4. How do you find the perpendicular distance from a point to a plane in 3D space?
5. Can you find the distance between two parallel lines?
6. How does this method change for non-linear equations?
7. What are some applications of point-to-line distance calculations in real life?
8. How would you find the shortest distance between two skew lines in space?

**Tip:** Always double-check your values when substituting into formulas to avoid simple arithmetic errors.

Learn how to calculate the distance from the point (2, 1) to the line 4x - 3y + 5 = 0 using the distance formula. This problem involves finding the perpendicular distance in a geometric plane, suitable for high school students studying geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation