line l contains points (0,-3) and (7,4) point p has coordinates (4,3) find the distance between point p and line l using the distance formula in geometry

To find the distance between point \( P(4, 3) \) and the line \( l \) passing through points \( A(0, -3) \) and \( B(7, 4) \), we can use the **point-to-line distance formula**:

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

Where:
- \( A, B, \) and \( C \) are the coefficients of the line equation \( Ax + By + C = 0 \),
- \( (x_1, y_1) \) are the coordinates of point \( P \).

### Step 1: Find the equation of the line \( l \).

The slope of the line passing through \( A(0, -3) \) and \( B(7, 4) \) is:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-3)}{7 - 0} = \frac{7}{7} = 1
\]

The equation of a line in point-slope form is:

\[
y - y_1 = m(x - x_1)
\]

Using point \( A(0, -3) \) and slope \( m = 1 \):

\[
y - (-3) = 1(x - 0)
\]

\[
y + 3 = x \quad \text{or} \quad x - y - 3 = 0
\]

Thus, the equation of the line is:

\[
x - y - 3 = 0
\]

### Step 2: Apply the point-to-line distance formula.

In the equation \( x - y - 3 = 0 \), we have \( A = 1 \), \( B = -1 \), and \( C = -3 \). Now, substitute the coordinates of point \( P(4, 3) \) into the formula:

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

\[
d = \frac{|4 - 3 - 3|}{\sqrt{1 + 1}} = \frac{|4 - 6|}{\sqrt{2}} = \frac{|-2|}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}
\]

So, the distance between point \( P(4, 3) \) and the line \( l \) is:

\[
d = \sqrt{2}
\]

### Conclusion:

The distance between point \( P(4, 3) \) and the line \( l \) is \( \sqrt{2} \) units.

---

Would you like more details or further clarification? Here are some related questions for deeper understanding:

1. How do you derive the distance formula between a point and a line?
2. What is the geometric interpretation of the slope in a line equation?
3. How would the distance change if point \( P \) were at a different position, such as \( (1, 1) \)?
4. How can the distance formula be applied in three dimensions?
5. What are some practical applications of the point-to-line distance formula in physics or engineering?

**Tip:** Always double-check signs when working with distances to avoid errors, especially in the absolute value step!

Line l contains points (0,-3) and (7,4). Point P has coordinates (4,3). Find the distance between point P and line l using the distance formula in geometry.

Learn how to find the distance between a point and a line using the distance formula in geometry. This problem involves calculating the distance from point P(4, 3) to line l passing through points (0, -3) and (7, 4). Suitable for students in Grades 10-12, this solution covers key concepts in analytic geometry.

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