Coordinate Geometry: Finding Points And Distances

by ADMIN 50 views

Hey guys! Let's dive into a fun problem involving coordinate geometry. We’re given two points, F(-3, 7) and L(4, 5), and we need to find the coordinate of a point D that is opposite to point F. After that, we'll plot these points on a coordinate line and calculate the distance between points L and D. Sounds like a plan? Let's get started!

Finding the Coordinate of Point D

So, the first part of our problem is to find the coordinate of point D, which is opposite to point F(-3, 7). When we say "opposite" in this context, we're talking about reflecting the point across the origin (0, 0). Think of it like flipping the signs of both the x and y coordinates. Essentially, we want to find a point that is equidistant from the origin but in the opposite direction.

To find the coordinates of point D, we simply take the negative of both coordinates of point F. Point F has coordinates (-3, 7). So, to find the coordinates of point D, we do the following:

  • The x-coordinate of D is the negative of the x-coordinate of F: -(-3) = 3
  • The y-coordinate of D is the negative of the y-coordinate of F: -(7) = -7

Therefore, the coordinates of point D are (3, -7). It's like we've taken point F and mirrored it through the origin. This means that point D is exactly the same distance from the origin as point F, but on the opposite side. Understanding this concept of reflection and how coordinates change is super useful in various geometry problems. We've nailed the first part – finding point D! Let’s move on to plotting these points.

Plotting Points F, L, and D on a Coordinate Line

Alright, now let’s visualize these points! We need to plot points F(-3, 7), L(4, 5), and D(3, -7) on a coordinate line. Now, since we are dealing with two-dimensional coordinates, technically we need a coordinate plane (both x and y axes) to plot these points accurately. However, for the sake of simplicity and if the question implies a single coordinate line, we can project these points onto either the x-axis or y-axis. Let’s project them onto the x-axis for this explanation.

When projecting onto the x-axis, we only consider the x-coordinates of the points. This means we'll plot -3 for point F, 4 for point L, and 3 for point D. Here’s how you can imagine it:

  1. Draw a number line: Start by drawing a straight line. Mark the origin (0) in the middle. This is our reference point.
  2. Mark the points:
    • Point F: Find -3 on the number line and mark it as point F. Remember, it’s to the left of the origin because it’s negative.
    • Point L: Find 4 on the number line and mark it as point L. This is to the right of the origin since it’s positive.
    • Point D: Find 3 on the number line and mark it as point D. This is also to the right of the origin.

So, on our x-axis projection, point F is to the left of the origin, while points L and D are to the right. The order from left to right would be F, then D, then L. Visualizing this helps understand their relative positions. If we were to project onto the y-axis instead, we would use the y-coordinates (7, 5, and -7) and the points would be arranged along the y-axis accordingly. Remember, plotting points accurately helps in understanding spatial relationships, which is crucial in geometry! Now, let's move on to the final part: finding the distance between points L and D.

Finding the Distance from Point L to Point D

Okay, guys, the last part of our problem is to find the distance from point L(4, 5) to point D(3, -7). To do this, we’ll use the distance formula, which is derived from the Pythagorean theorem. The distance formula helps us find the straight-line distance between any two points in a coordinate plane.

The distance formula is:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Where:

  • (x₁, y₁) are the coordinates of the first point (in our case, point L)
  • (x₂, y₂) are the coordinates of the second point (in our case, point D)
  • d is the distance between the two points

Let's plug in the coordinates of points L(4, 5) and D(3, -7) into the formula:

  • x₁ = 4, y₁ = 5
  • x₂ = 3, y₂ = -7

So, the formula becomes:

d = √((3 - 4)² + (-7 - 5)²)

Now, let’s simplify step by step:

  1. Calculate the differences inside the parentheses:
    • (3 - 4) = -1
    • (-7 - 5) = -12
  2. Square these differences:
    • (-1)² = 1
    • (-12)² = 144
  3. Add the squared differences:
    • 1 + 144 = 145
  4. Take the square root of the sum:
    • d = √145

So, the distance from point L to point D is √145. This is approximately 12.04 units. Therefore, we've successfully found the distance between the two points using the distance formula. Remember, this formula is super handy for solving a variety of geometry problems involving distances between points. Great job, team! We've tackled all parts of the problem.

In summary, we found the coordinates of point D, plotted the points on a coordinate line (or rather, projected them onto the x-axis), and calculated the distance between points L and D. Keep practicing these types of problems, and you'll become a coordinate geometry pro in no time!