line l has equation y=x+3. find distance between l and point R(-2,-6)

To find the distance between the point \( R(-2, -6) \) and the line \( l: y = x + 3 \), we can use the formula for the distance between a point \( (x_1, y_1) \) and a line in the form \( Ax + By + C = 0 \):

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

First, we rewrite the equation of the line \( y = x + 3 \) in the general form \( Ax + By + C = 0 \):

\[
x - y + 3 = 0
\]

Here, \( A = 1 \), \( B = -1 \), and \( C = 3 \).

Now, apply the point \( R(-2, -6) \) into the distance formula:

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

Simplifying:

\[
d = \frac{|-2 + 6 + 3|}{\sqrt{1 + 1}} = \frac{|7|}{\sqrt{2}} = \frac{7}{\sqrt{2}} = \frac{7\sqrt{2}}{2}
\]

Thus, the distance between the point \( R(-2, -6) \) and the line \( l \) is \( \frac{7\sqrt{2}}{2} \) units.

Would you like more details, or have any questions?

Here are 5 related questions to explore:

1. How do you find the perpendicular distance between a point and a plane in 3D?
2. What is the geometric interpretation of the distance between a point and a line?
3. How would the formula change for a vertical line like \( x = 5 \)?
4. How can you derive the distance formula for any two points in a plane?
5. What is the distance between the point \( (2, 3) \) and the line \( y = -x + 5 \)?

**Tip:** Always make sure the equation of the line is in standard form \( Ax + By + C = 0 \) before applying the distance formula.

Line l has equation y = x + 3. Find distance between l and point R(-2,-6).

Learn how to calculate the distance between the point R(-2, -6) and the line y = x + 3 using the distance formula. This problem involves key concepts in coordinate geometry, including converting a line equation into standard form and applying the point-line distance formula. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation