line l has equation y=-x+3. find the distance between l and point D(-4,-4) please be right

To find the distance between a point and a line, we use the formula:

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

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

In your case, the equation of the line \(l\) is given as \(y = -x + 3\), which can be rewritten in the standard form \(x + y - 3 = 0\). Therefore:
- \(A = 1\)
- \(B = 1\)
- \(C = -3\)

The point \(D\) has coordinates \((-4, -4)\), so \(x_1 = -4\) and \(y_1 = -4\).

Now, apply the values to the distance formula:

\[
d = \frac{|(1)(-4) + (1)(-4) - 3|}{\sqrt{1^2 + 1^2}} = \frac{| -4 - 4 - 3 |}{\sqrt{2}} = \frac{| -11 |}{\sqrt{2}} = \frac{11}{\sqrt{2}}
\]

Simplify the result:

\[
d = \frac{11\sqrt{2}}{2} \approx 7.78
\]

Thus, the distance between the line \(l\) and the point \(D(-4, -4)\) is approximately **7.78 units**.

Would you like more details or have any questions? Here are some related problems:

1. Find the distance between \(y = 2x - 5\) and point \(P(3, 7)\).
2. What is the distance between the line \(x - 2y + 4 = 0\) and the point \((1, -3)\)?
3. Determine the shortest distance from the origin to the line \(y = 3x + 2\).
4. Find the equation of the line parallel to \(y = -x + 3\) and passing through \((2, -5)\).
5. Calculate the perpendicular distance between the point \((5, 6)\) and the line \(3x + 4y - 10 = 0\).

**Tip**: Always rewrite line equations in the standard form \(Ax + By + C = 0\) to simplify calculations like distance or intersections!

Learn how to find the distance between the line y=-x+3 and the point D(-4, -4) using the distance formula. This problem covers important concepts in analytic geometry and distance calculations, suitable for students in Grades 10-12. Step-by-step solution included with formulas and key details.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation