Finding The Length Of A Horizontal Line Segment: What's The Next Step?
Leora is on an exciting mathematical journey! She's determined to find the length of a horizontal line segment gracefully placed on a coordinate grid. This task involves understanding the fundamental concepts of coordinate geometry and how distances are calculated in this mathematical landscape. Let's delve into Leora's challenge and discover the crucial step she needs to take next.
Understanding the Coordinate Grid
Before we jump into the specifics of Leora's problem, let's take a moment to appreciate the coordinate grid itself. Think of it as a map for numbers, a way to pinpoint locations with precision. The grid is formed by two perpendicular lines: the x-axis, which runs horizontally, and the y-axis, which runs vertically. Their meeting point, the origin, is the heart of the grid, the (0, 0) point from which all other locations are measured.
Any point on this grid can be uniquely identified by an ordered pair of numbers, (x, y), known as its coordinates. The x-coordinate tells us how far to move horizontally from the origin, and the y-coordinate tells us how far to move vertically. This elegant system allows us to represent geometric shapes and analyze their properties using the language of algebra.
When we talk about a horizontal line segment, we're referring to a straight line that runs parallel to the x-axis. This means that all points on this line share the same y-coordinate. The length of this segment is simply the distance between its endpoints along the x-axis.
Leora's Subtraction Adventure
Now, let's rejoin Leora in her quest. She's wisely chosen to subtract the x-coordinates of the endpoints of her horizontal line segment. This is a crucial first step, as the difference in x-coordinates directly relates to the horizontal distance between the points. She performed this subtraction and arrived at a value of -17.5. This is where her question arises: What should she do next?
The negative sign is a key piece of information. It tells us the direction of the distance but not the magnitude. Distance, in its essence, is a measure of length, and length cannot be negative. It's the same as saying you can't walk -5 meters; you can only walk 5 meters in the opposite direction. Therefore, Leora needs to transform this negative value into a positive one to accurately represent the length of the line segment.
The Absolute Value Solution
Here's where the concept of absolute value comes to the rescue. The absolute value of a number is its distance from zero, regardless of direction. Mathematically, it's represented by two vertical bars surrounding the number, like this: |x|. The absolute value of a positive number is the number itself, and the absolute value of a negative number is its positive counterpart. For instance, |5| = 5 and |-5| = 5.
Leora's next step, therefore, is to take the absolute value of -17.5. This will effectively remove the negative sign and give her the numerical length of the line segment. So, |-17.5| = 17.5. This means the line segment is 17.5 units long.
Why Not Add 17.5?
Option A, adding 17.5, might seem tempting, but it's not the correct approach. Adding 17.5 to -17.5 would result in zero, which doesn't represent the length of the line segment. Remember, we're interested in the magnitude of the distance, not its position relative to zero.
The Irrelevance of Y-Coordinates
Option C, adding the y-coordinates, is also incorrect. Since the line segment is horizontal, the y-coordinates of its endpoints are the same. Adding them would give us twice the y-coordinate value, which has no bearing on the length of the horizontal line segment. The length is solely determined by the difference in x-coordinates.
Leora's Triumphant Step
Leora's journey highlights the importance of understanding the underlying concepts in coordinate geometry. She correctly subtracted the x-coordinates, recognized the significance of the negative sign, and now knows that her next step is to take the absolute value of -17.5. This will give her the accurate length of the horizontal line segment, marking a successful step in her mathematical exploration.
By understanding the coordinate grid, the meaning of coordinates, and the concept of absolute value, Leora is well-equipped to tackle more complex geometric challenges in the future. Her journey is a testament to the power of mathematical reasoning and the satisfaction of finding the right solution.
Conclusion
In conclusion, when finding the length of a horizontal line segment on a coordinate grid, subtracting the x-coordinates is a crucial first step. However, if the result is negative, the next essential step is to take the absolute value of the result. This ensures that the length is represented as a positive value, accurately reflecting the distance between the endpoints of the line segment. Leora's problem serves as a valuable reminder of the importance of understanding mathematical concepts and applying them logically to solve real-world problems.
What is Leora's next step if she subtracted the x coordinates of a horizontal line segment and got -17.5?
Finding Length of Horizontal Line Segment What's Next Step?
