Problem 26
Question
Find the midpoint of each line segment with the given endpoints. $$\left(-\frac{2}{5}, \frac{7}{15}\right)\( and \)\left(-\frac{2}{5},-\frac{4}{15}\right)$$
Step-by-Step Solution
Verified Answer
So the midpoint of the line segment with the given endpoints is \((-\frac{2}{5}, \frac{1}{10})\).
1Step 1: Understanding the Midpoint formula
The formula to find the midpoint of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\)
2Step 2: Substituting Values into Formula
The given endpoints are \((-\frac{2}{5}, \frac{7}{15})\) and \((-\frac{2}{5},-\frac{4}{15})\). Substitute these values into the midpoint formula, giving:\(\frac{-\frac{2}{5} -\frac{2}{5}}{2}, \frac{\frac{7}{15} -\frac{4}{15}}{2}\)
3Step 3: Simplify
Doing the math yields the coordinates of the midpoint. -\frac{2}{5} - \frac{2}{5} = -\frac{4}{5} and divide this by 2 gives -\frac{2}{5}.\frac{7}{15} - \frac{4}{15} = \frac{3}{15} and divide this by 2 gives \frac{1}{10}.
Key Concepts
Line SegmentCoordinate GeometryAlgebra
Line Segment
A line segment is a part of a line that is defined by two endpoints. Unlike a line, which extends infinitely in both directions, a line segment has a fixed length. It connects two points in space—a beginning and an end. This concept is crucial in understanding geometric figures and their properties.
In mathematics, particularly in geometry, line segments are often labeled with the coordinates of their endpoints. This allows for precise descriptions of their position and length. For example, the endpoints \((-\frac{2}{5}, \frac{7}{15})\) and \((-\frac{2}{5}, -\frac{4}{15})\) create a vertical line segment on a plane. Understanding the positioning of such endpoints helps in calculating measurements like the length and the midpoint of the segment.
Working with line segments helps build foundational skills in math, which are essential in advanced topics such as calculus and physics. By dissecting and analyzing line segments, students gain a better grasp of mathematical relationships and spatial reasoning.
In mathematics, particularly in geometry, line segments are often labeled with the coordinates of their endpoints. This allows for precise descriptions of their position and length. For example, the endpoints \((-\frac{2}{5}, \frac{7}{15})\) and \((-\frac{2}{5}, -\frac{4}{15})\) create a vertical line segment on a plane. Understanding the positioning of such endpoints helps in calculating measurements like the length and the midpoint of the segment.
Working with line segments helps build foundational skills in math, which are essential in advanced topics such as calculus and physics. By dissecting and analyzing line segments, students gain a better grasp of mathematical relationships and spatial reasoning.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This branch of mathematics is instrumental in bringing algebra and geometry together, allowing us to represent geometric figures in a numerical way.
In a coordinate plane, every point can be identified by an ordered pair of numbers, known as coordinates. The first number in the pair refers to the point's horizontal position, while the second refers to its vertical position. For example, the endpoints \((-\frac{2}{5}, \frac{7}{15})\) and \((-\frac{2}{5}, -\frac{4}{15})\) pinpoint specific locations on the graph.
Understanding coordinate geometry is crucial for solving problems related to distance, midpoint, and slope of lines. The midpoint formula, \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\), is used to find the center point of a line segment by averaging the coordinates of the endpoints. This logical use of coordinates simplifies many complex geometrical problems, making it easier to visualize and solve.
In a coordinate plane, every point can be identified by an ordered pair of numbers, known as coordinates. The first number in the pair refers to the point's horizontal position, while the second refers to its vertical position. For example, the endpoints \((-\frac{2}{5}, \frac{7}{15})\) and \((-\frac{2}{5}, -\frac{4}{15})\) pinpoint specific locations on the graph.
Understanding coordinate geometry is crucial for solving problems related to distance, midpoint, and slope of lines. The midpoint formula, \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\), is used to find the center point of a line segment by averaging the coordinates of the endpoints. This logical use of coordinates simplifies many complex geometrical problems, making it easier to visualize and solve.
Algebra
Algebra is a branch of mathematics that uses symbols to represent numbers and express mathematical relationships. It is a universal tool for solving equations and understanding abstract concepts. Algebra simplifies many aspects of geometry by allowing us to use formulas rather than performing tedious repetition in calculations.
The midpoint formula is a perfect example of algebra in action: \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). This formula uses algebraic symbols to represent the process of finding the midpoint between two points. By using this formula, we can easily calculate the midpoint of the line segment with endpoints \((-\frac{2}{5}, \frac{7}{15})\) and \((-\frac{2}{5}, -\frac{4}{15})\).
Algebra also emphasizes the significance of simplifying expressions. To find the midpoint, we performed these calculations:
The midpoint formula is a perfect example of algebra in action: \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). This formula uses algebraic symbols to represent the process of finding the midpoint between two points. By using this formula, we can easily calculate the midpoint of the line segment with endpoints \((-\frac{2}{5}, \frac{7}{15})\) and \((-\frac{2}{5}, -\frac{4}{15})\).
Algebra also emphasizes the significance of simplifying expressions. To find the midpoint, we performed these calculations:
- \(-\frac{2}{5} - \frac{2}{5} = -\frac{4}{5}\) and divide by 2, resulting in \(-\frac{2}{5}\).
- \(\frac{7}{15} + \frac{4}{15} = \frac{3}{15}\) and divide by 2, resulting in \(\frac{1}{10}\).
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