how to find foot of perendicular at a line from a point

To find the foot of the perpendicular from a point to a line, we need to find the point on the line where a perpendicular drawn from the given point intersects the line.

Let's break down the steps:

### Given:
- A line in the form: \( Ax + By + C = 0 \)
- A point \( P(x_1, y_1) \)

### Steps:
1. **Find the slope of the given line:**
   - The slope \( m \) of the line \( Ax + By + C = 0 \) is \( -\frac{A}{B} \) (since in slope-intercept form, \( y = mx + c \)).

2. **Find the slope of the perpendicular line:**
   - The slope of the line perpendicular to the given line is the negative reciprocal of \( m \), which is \( \frac{B}{A} \).

3. **Write the equation of the perpendicular line:**
   - The equation of the line passing through \( P(x_1, y_1) \) with slope \( \frac{B}{A} \) is:
     \[
     y - y_1 = \frac{B}{A}(x - x_1)
     \]
     This can be rewritten as:
     \[
     A(y - y_1) = B(x - x_1)
     \]
     Simplifying, this gives:
     \[
     Ax - By = Ax_1 - By_1
     \]
     Now you have the equation of the perpendicular line.

4. **Solve for the intersection point:**
   - Solve the system of equations formed by:
     1. The original line \( Ax + By + C = 0 \)
     2. The perpendicular line \( Ax - By = Ax_1 - By_1 \)

     Solving these two equations simultaneously will give you the coordinates \( (x_2, y_2) \) of the foot of the perpendicular.

### Example:

Consider the line \( 3x + 4y - 5 = 0 \) and the point \( P(1, 2) \).

1. The slope of the given line is \( -\frac{3}{4} \).
2. The slope of the perpendicular line is \( \frac{4}{3} \).
3. The equation of the perpendicular line through \( P(1, 2) \) is:
   \[
   y - 2 = \frac{4}{3}(x - 1)
   \]
   Simplifying:
   \[
   y = \frac{4}{3}x - \frac{4}{3} + 2 = \frac{4}{3}x + \frac{2}{3}
   \]
4. Now, solve the system:
   \[
   3x + 4y - 5 = 0
   \]
   and
   \[
   y = \frac{4}{3}x + \frac{2}{3}
   \]
   Substituting the value of \( y \) into the first equation and solving for \( x \) and \( y \), you'll get the coordinates of the foot of the perpendicular.

Would you like further details on the solving step?

### Related Questions:
1. How do you derive the equation of a perpendicular bisector?
2. What is the general formula for finding the distance between a point and a line?
3. How do you find the equation of a line parallel to another line?
4. How can you find the intersection point of two lines?
5. How do you check if a point lies on a given line?

### Tip:
Always remember that the product of the slopes of two perpendicular lines is \( -1 \). This is a quick way to verify if two lines are perpendicular.

how to find foot of perpendicular at a line from a point

Learn how to find the foot of a perpendicular from a given point to a line with step-by-step guidance. This problem covers key concepts in coordinate geometry, including the calculation of slopes, equation of perpendicular lines, and solving systems of equations. Ideal for high school students in Grades 10-12.

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