Equation Of Median CM: Triangle ABC With Given Vertices

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Hey guys! Let's dive into a geometry problem where we need to find the equation of a median in a triangle. Specifically, we're looking for the equation of the median CM in triangle ABC, where we know the coordinates of the vertices: A(2; -3), B(8; -7), and C(4; 0). This might sound a bit daunting, but don't worry, we'll break it down step by step. So, grab your pencils and let's get started!

Understanding the Problem

First, let's make sure we're all on the same page. What exactly is a median in a triangle? Well, a median is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. In our case, CM is the median, which means it connects vertex C to the midpoint (let's call it M) of side AB. To find the equation of the line CM, we'll need two things: the coordinates of point C (which we already have) and the coordinates of point M. Once we have those, we can determine the slope of the line and use the point-slope form to write the equation. So, the initial key is to pinpoint the coordinates of point M, which sits perfectly in the middle of side AB. Stay with me, and we'll unravel this geometry puzzle together!

Finding the Midpoint M

Okay, let's find the coordinates of point M, the midpoint of side AB. Remember, the midpoint formula is our friend here. The midpoint formula states that if you have two points, say (x1, y1) and (x2, y2), the midpoint between them is simply ((x1 + x2)/2, (y1 + y2)/2). It's like finding the average of the x-coordinates and the average of the y-coordinates.

In our case, point A is (2, -3) and point B is (8, -7). So, let's plug these values into the midpoint formula:

  • The x-coordinate of M will be (2 + 8) / 2 = 10 / 2 = 5.
  • The y-coordinate of M will be (-3 + -7) / 2 = -10 / 2 = -5.

Therefore, the coordinates of point M are (5, -5). See? Not so bad, right? Now that we have the coordinates of both C (4, 0) and M (5, -5), we're one step closer to finding the equation of the median CM. Next up, we'll calculate the slope of this line. Keep those calculations coming!

Calculating the Slope of CM

Alright, now that we've nailed down the coordinates of points C (4, 0) and M (5, -5), let's figure out the slope of the line CM. The slope, often denoted as 'm', tells us how steep the line is and in what direction it's going. The formula for calculating the slope between two points (x1, y1) and (x2, y2) is: m = (y2 - y1) / (x2 - x1). It’s all about the change in y over the change in x – rise over run, as some might say!

Let’s plug in the coordinates of our points C and M into this formula. We'll let C be (x1, y1) which is (4, 0), and M be (x2, y2) which is (5, -5). So, here’s how the slope calculation looks:

  • m = (-5 - 0) / (5 - 4) = -5 / 1 = -5

So, the slope of the median CM is -5. That means for every one unit we move to the right along the line, we move five units down. A pretty steep line, indeed! With the slope in hand, we're now ready to roll out the point-slope form to find the equation of the line. Let’s keep the momentum going!

Using the Point-Slope Form

Okay, we've got the slope of the median CM, which is -5. Now, let's put that to good use with the point-slope form of a linear equation. This form is super handy when you know the slope of a line and a point it passes through. The point-slope form looks like this: y - y1 = m(x - x1), where 'm' is the slope, and (x1, y1) is the point.

We can use either point C (4, 0) or point M (5, -5) as our (x1, y1). Let's go with point C (4, 0) just to keep things simple. Now, we'll plug in the slope (-5) and the coordinates of point C into the point-slope form:

  • y - 0 = -5(x - 4)

See how it's all coming together? We're almost there! The next step is to simplify this equation and get it into the more familiar slope-intercept form (y = mx + b) or the standard form (Ax + By = C). Let's do that next and wrap this problem up!

Converting to Slope-Intercept Form

Fantastic! We've got the equation of the median CM in point-slope form: y - 0 = -5(x - 4). Now, let's make it look a little more polished by converting it into slope-intercept form, which is y = mx + b. This form is great because it clearly shows the slope (m) and the y-intercept (b) of the line.

To convert, we just need to distribute the -5 on the right side and simplify:

  • y - 0 = -5(x - 4)
  • y = -5x + 20

And there we have it! The equation of the median CM in slope-intercept form is y = -5x + 20. This tells us that the line has a slope of -5 (which we already knew) and crosses the y-axis at the point (0, 20). We could also convert it to standard form if we prefer, but for now, the slope-intercept form gives us a clear picture of the line. So, we've successfully found the equation of the median CM. Great job, guys!

Standard Form of the Equation

Now that we've got our equation in slope-intercept form (y = -5x + 20), let's take it one step further and convert it to the standard form of a linear equation. The standard form looks like this: Ax + By = C, where A, B, and C are constants, and A is usually a positive integer. Some people prefer standard form because it's a neat and tidy way to represent linear equations, and it can be useful in certain situations, like when solving systems of equations.

To convert our equation y = -5x + 20 to standard form, we need to move the -5x term to the left side of the equation. We can do this by adding 5x to both sides:

  • y = -5x + 20
  • 5x + y = 20

Voila! We now have the equation of the median CM in standard form: 5x + y = 20. See how clean and symmetrical it looks? It's just another way to represent the same line, but in a different format. Whether you prefer slope-intercept form or standard form often comes down to personal preference or the specific context of the problem. Now we've got it in both forms, so we're covered! Let's recap what we've done and wrap things up.

Recapping the Solution

Okay, let's take a moment to recap the journey we've been on and celebrate our success! We started with the challenge of finding the equation of the median CM in triangle ABC, given the coordinates of the vertices A(2; -3), B(8; -7), and C(4; 0). It seemed like a multi-step problem, but we tackled it head-on, step by step.

Here’s a quick rundown of what we did:

  1. Found the midpoint M of side AB: We used the midpoint formula to determine that M is (5, -5).
  2. Calculated the slope of CM: Using the slope formula, we found the slope to be -5.
  3. Applied the point-slope form: We plugged the slope and the coordinates of point C into the point-slope form to get y - 0 = -5(x - 4).
  4. Converted to slope-intercept form: We simplified the equation to y = -5x + 20.
  5. Converted to standard form: We further transformed the equation to 5x + y = 20.

So, we've not only found the equation of the median CM, but we've also expressed it in two common forms: slope-intercept and standard. How awesome is that? Geometry problems like this might seem tricky at first, but by breaking them down into smaller, manageable steps, we can conquer them. Keep practicing, and you'll become a geometry whiz in no time! Great work, everyone!