Parallel Lines: Find Equation Of CD

by Luna Greco 36 views

Hey everyone! Let's dive into a fun geometry problem today. We're dealing with parallel lines, coordinates, and equations. Specifically, we're given that lines AB and CD are parallel, the coordinates of point A are (-2, 5), point B is at (6, 3), and point D sits at (8, 5). Our mission, should we choose to accept it, is to find the equation of line CD. Buckle up, because we're about to break this down step by step!

Understanding Parallel Lines and Slopes

So, what's the key to cracking this? It all boils down to understanding parallel lines. Remember, parallel lines are those that run side by side, never intersecting. And here's the golden rule: Parallel lines have the same slope. This is absolutely crucial! If we can figure out the slope of line AB, we automatically know the slope of line CD. Think of it like this: they're traveling on the same incline, just on different tracks.

Now, how do we find the slope? The slope, often denoted by 'm', tells us how steep a line is. It's the "rise over run," or the change in y divided by the change in x. Mathematically, if we have two points (x1, y1) and (x2, y2), the slope is calculated as:

m = (y2 - y1) / (x2 - x1)

Let's put this into action! We have points A (-2, 5) and B (6, 3). Let's plug these values into our slope formula:

m_AB = (3 - 5) / (6 - (-2)) m_AB = (-2) / (8) m_AB = -1/4

Boom! We've got the slope of line AB, which is -1/4. Since lines AB and CD are parallel lines, the slope of line CD (m_CD) is also -1/4. This is a major step forward. We've conquered the first hurdle and now have a vital piece of information.

Leveraging the Point-Slope Form

Okay, we know the slope of line CD. What's next? We need to find its equation. One of the most useful forms for this is the point-slope form of a linear equation. This form is super handy when you have a point on the line and the slope. It looks like this:

y - y1 = m(x - x1)

Where:

  • m is the slope
  • (x1, y1) is a point on the line

Lucky for us, we have both! We know m_CD = -1/4, and we're given point D (8, 5), which lies on line CD. Let's plug these values into the point-slope form:

y - 5 = (-1/4)(x - 8)

This equation represents line CD, but it's not in the most common form, which is the slope-intercept form (y = mx + b). Let's transform it!

Converting to Slope-Intercept Form

The slope-intercept form is the classic y = mx + b, where:

  • m is the slope
  • b is the y-intercept (the point where the line crosses the y-axis)

To get our equation into this form, we need to distribute and isolate y. Let's take our equation from the point-slope form:

y - 5 = (-1/4)(x - 8)

First, distribute the -1/4:

y - 5 = (-1/4)x + 2

Now, add 5 to both sides to isolate y:

y = (-1/4)x + 2 + 5 y = (-1/4)x + 7

And there we have it! The equation of line CD in slope-intercept form is y = (-1/4)x + 7. We've successfully navigated through the concepts of parallel lines, slopes, the point-slope form, and the slope-intercept form to arrive at our answer.

Putting it All Together: A Recap

Let's quickly recap the steps we took to solve this problem:

  1. Recognized that parallel lines have the same slope.
  2. Calculated the slope of line AB using the formula m = (y2 - y1) / (x2 - x1).
  3. Determined that the slope of line CD is the same as line AB.
  4. Used the point-slope form (y - y1 = m(x - x1)) with the slope of CD and point D.
  5. Converted the equation from point-slope form to slope-intercept form (y = mx + b) by distributing and isolating y.

By following these steps, we successfully found the equation of line CD. Geometry problems like these might seem daunting at first, but by breaking them down into smaller, manageable steps and understanding the underlying principles, you can conquer them with confidence. Keep practicing, and you'll be a geometry whiz in no time!

Final Thoughts

This problem highlights the importance of understanding fundamental geometric concepts and how they connect. The relationship between parallel lines and their slopes is a cornerstone of coordinate geometry. Being comfortable with different forms of linear equations, like the point-slope form and slope-intercept form, is also crucial for solving these types of problems. Remember, practice makes perfect! The more you work through these types of problems, the more intuitive they'll become. So, keep those pencils sharpened and keep exploring the fascinating world of geometry!

What is the equation of line CD given that lines AB and CD are parallel, the coordinates of point A are (-2,5), the coordinates of point B are (6,3), and the coordinates of point D are (8,5)?

Parallel Lines: Find Equation of Line CD