Solving Coordinate Geometry Problems: A Step-by-Step Guide

Hey everyone! Let's dive into some coordinate geometry problems together. Don't worry, it's not as scary as it sounds. We'll break down the questions step-by-step to make sure we understand everything. Coordinate geometry is all about using numbers to describe points, lines, and shapes. It's super useful for understanding the world around us. So, grab your pencils and let's get started. We'll tackle these problems like pros! We'll start by looking at which points belong to certain coordinate planes and axes. Then, we will understand how to identify points on the x-axis. Finally, we'll discuss the main concepts that will help solve these coordinate geometry problems.

Question 1: Identifying Points on the XZ Coordinate Plane

Coordinate Geometry is a fantastic tool that allows us to precisely locate points in space. Our first question asks us to determine which of the provided points – A(7;9;0), B(0;-6;8), or C(-4;0;5) – lies on the xz coordinate plane. This plane is defined by all points where the y-coordinate is equal to zero. Remember that, in a 3D coordinate system, we have three axes: the x-axis, the y-axis, and the z-axis. These axes are perpendicular to each other, and they intersect at a point called the origin (0;0;0). The xz-plane is like a flat sheet that includes both the x-axis and the z-axis, and it extends infinitely in all directions within that plane. So, when we're looking for a point on the xz-plane, we need to find the point where the y-coordinate is 0. Let's examine each point. The point A(7;9;0) has a y-coordinate of 9. Therefore, it does not belong to the xz-plane. The point B(0;-6;8) has a y-coordinate of -6, which means it also does not belong to the xz-plane. Finally, the point C(-4;0;5) has a y-coordinate of 0. This means it lies on the xz-plane! Therefore, the answer is point C.

Now, let's explore why this is the case. Imagine standing at the origin (0;0;0). To get to point A, you'd move 7 units along the x-axis, then 9 units along the y-axis, and finally, 0 units along the z-axis. Because you move along the y-axis, point A is not on the xz-plane. In contrast, to get to point C, you'd move -4 units along the x-axis, 0 units along the y-axis, and 5 units along the z-axis. Since you don't move at all along the y-axis, point C lies directly on the xz-plane. Visualizing the coordinate planes and the points in 3D space can be helpful when solving these types of problems. You can sketch a quick diagram to help you see where each point is located relative to the xz-plane. Always remember that any point on the xz-plane has a y-coordinate of 0. This is the key to solving this type of problem. So, when you come across similar questions, don’t hesitate to write the coordinates of the points and circle the y-coordinate. If that one is zero, then you've found your point! Keep practicing, and you'll become a coordinate geometry whiz!

Question 2: Identifying Points on the Ox Axis

Alright, let's move on to our second problem. This time, we need to identify which of the following points lies on the Ox axis. The Ox axis, also known as the x-axis, is a fundamental component of our coordinate system. In a 3D coordinate system, the x-axis is a straight line that passes through the origin. All points on the x-axis have a y-coordinate and a z-coordinate of zero. So, to solve this problem, we need to look at each point and determine if its y and z coordinates are both zero. Let's examine the options. The point A(1;0;1) has a y-coordinate of 0, but its z-coordinate is 1. Therefore, it is not on the Ox axis. The point B(0;0;4) has a y-coordinate of 0, but its x-coordinate is also 0. It means that point lies on the z-axis. Thus, it is not on the Ox axis. The point C(-1;0;0) has a y-coordinate of 0 and a z-coordinate of 0. This means it does lie on the Ox axis! Finally, the point D(1;2;0) has a z-coordinate of 0, but its y-coordinate is 2. Therefore, it is not on the Ox axis.

Now, let's think about why this is the case. Imagine the x-axis. It's a line that stretches infinitely in both positive and negative directions. Any point on this line can be described by its distance from the origin along the x-axis. Since the x-axis is a horizontal line, there is no vertical movement (y-coordinate) or movement into or out of the page (z-coordinate). Think of it like walking on a perfectly flat surface. You are only moving horizontally (x-coordinate). You are not moving up or down (y-coordinate) or towards or away from you (z-coordinate). So, any point on the x-axis has the form (x;0;0). The x value can be any real number, but the y and z values must always be 0. When you encounter similar problems, remember this rule: points on the x-axis have a y-coordinate and z-coordinate of zero. This is the key to solving this type of problem. To recap, to be on the x-axis, a point must have coordinates in the form (x;0;0). The correct answer is point C(-1;0;0). Keep practicing, and you'll master these types of questions with ease!

Discussion Category: Geometry

Let's move on to the discussion category: Geometry. Geometry is a fascinating branch of mathematics that explores the properties of space, shapes, sizes, and relative positions of figures. It's a field with ancient roots, dating back to the early civilizations of Egypt and Mesopotamia, who used geometric principles for practical purposes like surveying land and constructing buildings. Today, geometry is still a fundamental aspect of many fields, including architecture, engineering, computer graphics, and even art. Coordinate geometry, which we've been practicing, is a powerful tool within geometry that lets us represent geometric objects and solve problems using algebraic techniques. It allows us to apply the principles of algebra to geometric figures, providing a more versatile approach to problem-solving. It's essentially a way to bridge the gap between algebra and geometry, allowing us to combine two powerful mathematical tools. The basic idea of coordinate geometry is to use a coordinate system, like the Cartesian coordinate system, to assign numbers to points in space. These numbers, called coordinates, tell us the position of a point relative to a fixed reference point, usually the origin. These coordinates can then be used to calculate distances, areas, and other geometric properties.

Understanding the relationship between algebra and geometry is crucial. Think of an equation like y = 2x + 1. In algebra, this is a linear equation. But in coordinate geometry, this equation represents a straight line. Every point (x, y) that satisfies the equation lies on this line. This ability to visualize algebraic equations as geometric shapes opens up a whole new world of problem-solving possibilities. We can use algebraic tools to analyze geometric figures and vice versa. For example, we can use the distance formula to calculate the distance between two points, the midpoint formula to find the midpoint of a line segment, and the slope formula to determine the slope of a line. These formulas are all derived from geometric principles but are expressed in algebraic terms. In geometry, there are many types of figures, such as lines, angles, triangles, circles, and 3D shapes. Each of these figures has its own unique properties. Understanding these properties is crucial to solving geometric problems. Coordinate geometry provides a way to represent these figures with algebraic equations and then to manipulate these equations to solve problems. It's a powerful tool that combines the best aspects of algebra and geometry.

Mastering coordinate geometry requires practice. Solving problems like the ones we've covered, where you need to identify points on coordinate planes or axes, helps you build a strong foundation. Regularly working through examples and understanding the underlying concepts is the key to success. Don't be afraid to draw diagrams. Visualizing the problem can often make the solution clearer. Make sure to identify and understand the key concepts like the distance formula, the midpoint formula, and the slope of a line, as these will be essential tools. The more you practice, the more comfortable you'll become with this fascinating area of math.