Geometry Problems: Finding Points And Distances

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Hey guys! Let's dive into some geometry problems. We're going to explore how to find distances and points, which is super important in geometry. We'll be working with coordinate systems and applying some cool formulas. So, grab your pens and paper, and let's get started! We will address the question of the problem, including the coordinates of points and determine the distance between them. This is a step-by-step guide that will help you understand the process and solve similar problems with ease. We'll look at how to apply the distance formula and the concept of equidistance to tackle these challenges. These concepts are fundamental in geometry and are widely used in various fields, including computer graphics, physics, and engineering. Understanding them will not only help you in your studies but also provide a solid foundation for more advanced topics. The coordinate system helps us locate points in space. Each point is represented by a pair of numbers (x, y), where 'x' is the horizontal position and 'y' is the vertical position.

Problem 948: Distance Between Points

Understanding the Problem

Firstly, let's clarify the task. We are given three points: A, B, and C, and we need to find the distance between these points. The coordinates of the points change, so we will consider several cases. Remember that we will be using the distance formula, which is derived from the Pythagorean theorem. Let's understand what this formula tells us. The distance formula helps to calculate the distance between two points on a coordinate plane. The formula itself is pretty straightforward: d = sqrt((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of the two points, and 'd' represents the distance. This formula is the core of our problem-solving strategy. This formula is not just a bunch of numbers; it represents the length of the straight line between two points. The formula incorporates the idea of how far apart the points are horizontally and vertically, giving us the exact distance. The formula is derived from the Pythagorean theorem applied to the right-angled triangle formed by the points and their projections on the axes. The coordinates of the points are given in the problem statement. The distance formula is a fundamental tool in coordinate geometry. Using this formula will give the answer to the distance between the given points. This knowledge will make understanding the geometry problems easy. Using the distance formula, we can determine the distance between any two points. The formula is our tool to get the exact value.

Solving Part a: A(0, 1), B(1, -4), C(5, 2)

To calculate the distance between the points, we'll need to do it step-by-step. We will apply the distance formula. First, find the distance between points A and B. Then find the distance between points B and C. The problem gives us the coordinates for points A, B, and C, which are (0, 1), (1, -4), and (5, 2) respectively. Using the distance formula, we can calculate the distance between any two of these points. This process is repeated for each pair of points to determine the total distance.

  1. Distance AB: We have A(0, 1) and B(1, -4). Plugging the coordinates into the distance formula: d = sqrt((1 - 0)^2 + (-4 - 1)^2) = sqrt(1^2 + (-5)^2) = sqrt(1 + 25) = sqrt(26). The distance between A and B is √26.
  2. Distance BC: We have B(1, -4) and C(5, 2). Using the distance formula: d = sqrt((5 - 1)^2 + (2 - (-4))^2) = sqrt(4^2 + 6^2) = sqrt(16 + 36) = sqrt(52). The distance between B and C is √52.

So, for part (a), we've calculated the distances AB and BC.

Solving Part b: A(-4, 1), B(-2, 4), C(0, 1)

Now, let's tackle part (b). We will follow the same method as in part (a). This problem provides the coordinates of points A, B, and C, which are (-4, 1), (-2, 4), and (0, 1), respectively. We need to calculate the distance between each pair of points using the distance formula. The calculation is done step by step. For this part of the problem, we'll again calculate the distances between the points. The distance formula helps us find the distance between two points on a coordinate plane. We can understand how to calculate the distance between these points with the help of the distance formula. Applying the distance formula allows us to find the total distance between the points. The distance formula, when applied correctly, gives an accurate measure of the distance between two points. This method ensures that all pairs of points are covered in the calculation.

  1. Distance AB: With A(-4, 1) and B(-2, 4): d = sqrt((-2 - (-4))^2 + (4 - 1)^2) = sqrt(2^2 + 3^2) = sqrt(4 + 9) = sqrt(13). The distance between A and B is √13.
  2. Distance BC: With B(-2, 4) and C(0, 1): d = sqrt((0 - (-2))^2 + (1 - 4)^2) = sqrt(2^2 + (-3)^2) = sqrt(4 + 9) = sqrt(13). The distance between B and C is √13.

And there you go! We've calculated the distances for part (b).

Problem 949: Finding Equidistant Points on the Coordinate Axes

Understanding the Problem

In Problem 949, we are looking for points that are the same distance from two given points. In the first part, we will find a point on the y-axis that is equidistant from two other points. In the second part, we will find a point on the x-axis that is equidistant from two other points. This problem involves understanding the concept of equidistance. The concept of equidistance is crucial here. The objective is to find a point that maintains an equal distance to two other fixed points. When a point is equidistant from two other points, it means that the distances from that point to each of the other two points are equal. The key to solving this problem lies in understanding how to set up equations based on the distance formula and the conditions given. Our goal is to find the coordinates of a point on the y-axis or x-axis. So let's explore how to solve it.

Solving Part a: Equidistant Point on the Y-axis

The Problem

We need to find a point on the y-axis that is equidistant from A(-3, 5) and B(6, 4). Remember, any point on the y-axis has an x-coordinate of 0. So, our point will be (0, y). The y-axis is the vertical line where x equals zero. Therefore, to find the equidistant point, we need to determine the y-coordinate of the point. We are looking for a point on the y-axis (where x=0) that has equal distances to points A and B. Any point on the y-axis has coordinates (0, y). We need to set up an equation using the distance formula to solve for the y-coordinate.

The Solution

  1. Define the Point: Let the point on the y-axis be P(0, y).
  2. Apply the Distance Formula: We need to calculate the distance from P to A and P to B and set them equal. The distance PA = sqrt((-3 - 0)^2 + (5 - y)^2) and PB = sqrt((6 - 0)^2 + (4 - y)^2). Set PA = PB.
  3. Set up the Equation: sqrt((-3)^2 + (5 - y)^2) = sqrt(6^2 + (4 - y)^2). Square both sides to get rid of the square roots:
    • 9 + (25 - 10y + y^2) = 36 + (16 - 8y + y^2)
  4. Solve for y: Simplify the equation:
    • 34 - 10y + y^2 = 52 - 8y + y^2
    • -2y = 18
    • y = -9

So, the point on the y-axis equidistant from A and B is (0, -9).

Solving Part b: Equidistant Point on the X-axis

The Problem

We need to find a point on the x-axis that is equidistant from C(4, -3) and D(8, 1). Any point on the x-axis has a y-coordinate of 0. So, our point will be (x, 0). To solve this problem, you need to find the x-coordinate. When finding a point on the x-axis, the y-coordinate is always zero. Now we apply the distance formula to find the x-coordinate of this point. The problem gives us two points, C and D. We want to find a point on the x-axis that is equidistant from both C and D.

The Solution

  1. Define the Point: Let the point on the x-axis be Q(x, 0).
  2. Apply the Distance Formula: We need to calculate the distance from Q to C and Q to D and set them equal. The distance QC = sqrt((4 - x)^2 + (-3 - 0)^2) and QD = sqrt((8 - x)^2 + (1 - 0)^2). Set QC = QD.
  3. Set up the Equation: sqrt((4 - x)^2 + (-3)^2) = sqrt((8 - x)^2 + 1^2). Square both sides:
    • (16 - 8x + x^2) + 9 = (64 - 16x + x^2) + 1
  4. Solve for x: Simplify and solve:
    • 25 - 8x + x^2 = 65 - 16x + x^2
    • 8x = 40
    • x = 5

So, the point on the x-axis equidistant from C and D is (5, 0).

Conclusion

Awesome, guys! We've successfully navigated through these geometry problems. We've learned how to find distances between points using the distance formula and how to identify points equidistant from others on the coordinate axes. Remember that practice is key. Keep solving problems, and you'll become a geometry pro in no time. Keep practicing and you'll ace these problems with ease! Hopefully, this helped you understand the basics of these kinds of problems. See you in the next lesson!