Given The Coordinates Below, What Is AB?A(-6, - 10)B(2,5)A:5B:25C:17D:7
Understanding how to determine the length of a segment between two points on a coordinate plane is a fundamental skill in geometry and mathematics. In this article, we will explore the problem of calculating the length of segment AB given the coordinates of points A and B, analyze the provided options, and walk through the step-by-step process to arrive at the correct answer. Whether you're a student preparing for exams or someone interested in coordinate geometry, this comprehensive guide will clarify the concepts involved, provide detailed calculations, and answer common questions related to the topic.
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Understanding Coordinates and the Distance Formula
What Are Coordinates in Geometry?
In coordinate geometry, each point in the plane is represented by an ordered pair of numbers, called coordinates. These coordinates specify the point's position relative to two perpendicular axes:
- The x-axis (horizontal axis)
- The y-axis (vertical axis)
A point \(A\) with coordinates \((x1, y1)\) indicates it is located \(x1\) units along the x-axis and \(y1\) units along the y-axis.
For example, in the given problem:
- Point \(A\) has coordinates \((-6, -10)\)
- Point \(B\) has coordinates \((2, 5)\)
The Distance Formula
To find the length of the segment \(AB\), we use the distance formula derived from the Pythagorean theorem:
\[
\text{Distance } AB = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}
\]
Where:
- \((x1, y1)\) are the coordinates of point \(A\)
- \((x2, y2)\) are the coordinates of point \(B\)
This formula calculates the straight-line (Euclidean) distance between two points in the coordinate plane.
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Step-by-Step Calculation of AB
Identify Coordinates
Given:
- \(A(-6, -10)\)
- \(B(2, 5)\)
Compute Differences in Coordinates
Calculate the differences:
- \(\Delta x = x2 - x1 = 2 - (-6) = 2 + 6 = 8\)
- \(\Delta y = y2 - y1 = 5 - (-10) = 5 + 10 = 15\)
Apply the Distance Formula
Plug into the formula:
\[
AB = \sqrt{(8)^2 + (15)^2} = \sqrt{64 + 225} = \sqrt{289}
\]
Calculate the Square Root
\[
AB = \sqrt{289} = 17
\]
Thus, the length of segment \(AB\) is 17 units.
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Analyzing the Multiple-Choice Options
The problem provides the following options:
- A: 5
- B: 25
- C: 17
- D: 7
Based on our calculation, the correct answer is C: 17.
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Additional Insights and Related Concepts
Why Is the Distance Formula Important?
The distance formula is fundamental in various fields such as geometry, physics, engineering, and computer graphics. It helps determine lengths, distances between points, and is crucial in proofs and problem-solving involving coordinate planes.
Other Applications of Coordinates and Distance Calculation
- Finding the length of diagonals in polygons
- Calculating the shortest path between two points
- Determining the radius of circles when points are given
- Analyzing geometric figures in coordinate systems
Common Mistakes to Avoid
- Mixing up the coordinates of points \(A\) and \(B\)
- Forgetting to square the differences
- Not taking the square root after summing squares
- Confusing the order of subtraction (though it doesn't affect the result due to squaring)
Additional Practice Problems
Engaging with similar problems can reinforce understanding. Here are some practice questions:
- Find the distance between points \(C(3, -4)\) and \(D(-1, 2)\).
- Given points \(E(0, 0)\) and \(F(6, 8)\), what is the length of segment \(EF\)?
- If point \(G(x, y)\) is at \((-3, 7)\) and point \(H(4, -2)\), what is the distance between \(G\) and \(H\)?
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Summary and Key Takeaways
- The coordinates of points are essential in calculating distances in coordinate geometry.
- The distance formula is derived from the Pythagorean theorem and used to find the length of segments between two points.
- For the points \(A(-6, -10)\) and \(B(2, 5)\), the length of segment \(AB\) is 17 units.
- Accurate calculation involves careful handling of differences in x and y coordinates, squaring, summing, and square-rooting.
Conclusion
Understanding how to calculate the distance between two points using their coordinates is a vital skill in mathematics. The problem posed—finding the length of segment \(AB\) given specific coordinates—demonstrates how straightforward applying the distance formula can yield an exact answer. Remember, practice is key; working through similar problems will strengthen your grasp of coordinate geometry and prepare you for more complex applications.
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Meta Description:
Learn how to calculate the length of segment AB given points A(-6, -10) and B(2, 5). Step-by-step guide to the distance formula, solving multiple-choice options, and related coordinate geometry concepts.
