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Magazine Chapter 12: Problem 72
Find the distance from the given point to the horizontal axis. $$(-5,1)$$
Short Answer
Expert verified
The distance from the point (-5,1) to the horizontal axis is 1.
Step by step solution
01
Identify the Given Point
The given point on the coordinate plane is (-5,1). With this point, the first number (-5) is the x-coordinate and the second number (1) is the y-coordinate.
02
Understand the Distance to the Horizontal Axis
The horizontal axis on a two-dimensional (2-D) coordinate plane is the x-axis, which runs from left to right. Any point on the x-axis has the y-coordinate equal to zero. Thus, the distance of a point to the x-axis equals the absolute value of its y-coordinate.
03
Calculate the Distance
The y-coordinate of the given point (-5,1) is 1. The distance to the horizontal axis is hence equal to the absolute value of the y-coordinate which is \( |1| = 1 \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coordinate Plane
The concept of a coordinate plane is fundamental in understanding how positions and distances are visualized in mathematics. A coordinate plane is a two-dimensional plane formed by two perpendicular lines: the horizontal axis, called the x-axis, and the vertical axis, called the y-axis. These axes divide the plane into four sections known as quadrants. Each point on this plane can be uniquely identified by a pair of numbers: the x-coordinate and the y-coordinate.
- The x-coordinate tells how far to move horizontally from the origin, which is the point (0,0) where the axes intersect.
- The y-coordinate indicates how far to move vertically.
Distance Calculation
Calculating distance on a coordinate plane often involves finding how far apart two points are. However, in this exercise, we are specifically finding the distance from a point to the horizontal axis, which simplifies the process.
- For a given point \((x, y)\), the distance to the x-axis is essentially the length of the vertical line segment connecting the point to the x-axis.
- This is done by focusing solely on the y-coordinate, as it represents the vertical shift from the x-axis.
Horizontal Axis
The horizontal axis, commonly referred to in a coordinate plane context as the x-axis, serves as the baseline reference for horizontal positioning. It runs from left to right across the plane and is crucial for the two-dimensional mapping of coordinates. The horizontal axis helps in separating different quadrants on the coordinate plane.
- It is always at y = 0.
- Knowing this aids in understanding why the distance from a point to the x-axis can be derived from the y-coordinate.
X-axis
The x-axis plays a crucial role in any coordinate graphing system, acting as the horizontal reference line. It is the axis of orientation where the value of y is zero. This makes it an essential point of reference in understanding spatial relationships and distance calculations.
- Every point along the x-axis has a y-coordinate of zero.
- This feature simplifies distance calculations since the distance of any given point \((x, y)\) to the x-axis depends directly on how far the y-value is from zero.
Absolute Value
The concept of absolute value is crucial in mathematics for determining distance. The absolute value of a number is its distance from zero on the number line, regardless of direction. It's denoted as \(|y|\) for a number \y\. This is because distance can never be negative.
- In the context of coordinate planes, the distance from a point to the x-axis is the absolute value of its y-coordinate.
- This represents how far the point is vertically from the baseline \(y = 0\).
