Glossary

13.1 Distance and midpoint formulas

3 min read•july 22, 2024

The distance and midpoint formulas are key tools for working with line segments in a coordinate plane. They help you find lengths between and locate the center of a segment, which is super useful in geometry.

These formulas are based on the Pythagorean theorem and simple averages. You'll use them to solve real-world problems, like finding distances between cities or midpoints of bridges. Mastering these will make many geometry tasks much easier.

Distance and Midpoint Formulas

Distance formula for line segments

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  • Calculates the of a between two points in a coordinate plane
  • Derived from the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) by considering the line segment as the hypotenuse of a right triangle
  • Formula: For points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), the distance dd between them is d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
  • To find the length of a line segment:
    • Identify the of the endpoints (start and end points)
    • Substitute the xx and yy values of the endpoints into the distance formula
    • Simplify the expression under the square root by squaring the differences and adding them together
    • Calculate the square root to find the final distance between the points
  • Example: Find the distance between points (3,4)(3, 4) and (7,1)(7, 1)
    • d=(7−3)2+(1−4)2=42+(−3)2=16+9=25=5d = \sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5

Midpoint formula for coordinates

  • Determines the coordinates of the point that divides a line segment into two equal parts
  • Calculates the of the xx-coordinates and yy-coordinates of the endpoints
  • Formula: For endpoints (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), the midpoint MM is M=(x1+x22,y1+y22)M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})
  • To find the midpoint:
    • Identify the coordinates of the endpoints of the line segment
    • Add the xx-coordinates of the endpoints and divide the sum by 2 to find the xx-coordinate of the midpoint
    • Add the yy-coordinates of the endpoints and divide the sum by 2 to find the yy-coordinate of the midpoint
    • Write the midpoint as an ordered pair (x,y)(x, y)
  • Example: Find the midpoint of the line segment with endpoints (−2,5)(-2, 5) and (6,9)(6, 9)
    • xx-coordinate of midpoint: −2+62=42=2\frac{-2 + 6}{2} = \frac{4}{2} = 2
    • yy-coordinate of midpoint: 5+92=142=7\frac{5 + 9}{2} = \frac{14}{2} = 7
    • Midpoint: (2,7)(2, 7)

Applications of distance and midpoint

  • Recognize when to use the distance or in real-world problems
  • Identify relevant information such as coordinates of points or lengths of segments
  • Apply the appropriate formula to solve the problem:
    • Distance formula for finding lengths or distances between points (cities, landmarks)
    • Midpoint formula for finding the center point of a line segment (roads, bridges)
  • Interpret the results in the context of the problem considering units of measurement (miles, kilometers)
  • Example: Two cities are located at (120,250)(120, 250) and (400,150)(400, 150) on a map with a scale of 1 unit = 10 miles. Find the distance between the cities.
    • Coordinates: (120,250)(120, 250) and (400,150)(400, 150)
    • d=(400−120)2+(150−250)2=2802+(−100)2=78400+10000=88400≈297d = \sqrt{(400 - 120)^2 + (150 - 250)^2} = \sqrt{280^2 + (-100)^2} = \sqrt{78400 + 10000} = \sqrt{88400} \approx 297 units
    • Scale: 1 unit = 10 miles, so the actual distance is 297×10≈2970297 \times 10 \approx 2970 miles

Key Terms to Review (13)

(x1, y1): The notation (x1, y1) represents the coordinates of a point in a two-dimensional Cartesian coordinate system. Each coordinate specifies the position of the point relative to the x-axis and y-axis, where 'x1' indicates the horizontal distance from the origin and 'y1' indicates the vertical distance. Understanding this notation is essential for calculating distances between points and finding midpoints, which are foundational concepts in geometry.
(x2, y2): The notation (x2, y2) represents the coordinates of a point in a two-dimensional Cartesian coordinate system, where 'x2' indicates the horizontal position and 'y2' indicates the vertical position. These coordinates are essential for understanding geometric concepts such as distance and midpoint, which rely on the precise location of points in a plane. Knowing how to interpret and manipulate these coordinates is crucial for solving problems related to distances between points or finding midpoints along line segments.
Average: Average is a central value that summarizes a set of numbers, often calculated to represent the typical or expected value within a data set. In geometry, particularly when discussing distance and midpoint formulas, average helps in finding the central point between two coordinates, thus aiding in understanding spatial relationships.
Calculating Midpoints: Calculating midpoints involves finding the point that is exactly halfway between two endpoints in a coordinate system. This concept is essential in geometry as it helps in understanding distance, symmetry, and the division of segments into equal parts. The midpoint serves as a crucial reference point for various geometric constructions and analyses.
Coordinates: Coordinates are a set of numerical values that define the position of a point in a space, typically expressed in terms of its distance from reference axes. They provide a way to locate points in a two-dimensional or three-dimensional system, enabling calculations related to distance and midpoints. In geometry, coordinates are fundamental for analyzing shapes, distances, and positions in a clear and precise manner.
Distance between two points: The distance between two points in a coordinate plane is a measure of the straight line length that connects those points. This concept is fundamental in geometry, allowing for the calculation of how far apart two locations are in a two-dimensional space. It can be determined using specific formulas that incorporate the coordinates of the points involved.
Euclidean Space: Euclidean space is a fundamental concept in geometry that describes a two-dimensional or three-dimensional space characterized by the familiar geometric properties of points, lines, and planes. It provides the framework for understanding shapes and distances, where the axioms of Euclidean geometry, such as the parallel postulate, hold true. This type of space is essential for applying distance and midpoint formulas, which help in determining measurements between points in such environments.
Finding distance: Finding distance refers to calculating the space between two points in a coordinate system, typically using mathematical formulas that derive from geometric principles. It plays a critical role in understanding spatial relationships and is essential for determining lengths in various contexts, such as navigation, geometry, and physics. The concept encompasses techniques like the distance formula and the application of the Pythagorean theorem, which link geometry with algebra for practical problem-solving.
Length: Length refers to the measurement of how long an object or a segment is, usually expressed in units such as inches, feet, centimeters, or meters. It is a fundamental concept in geometry that allows for the comparison and calculation of distances between points, which plays a crucial role in understanding ratios, proportions, and geometric properties.
Line segment: A line segment is a part of a line that is bounded by two distinct endpoints. It represents the shortest distance between those two points and is a fundamental concept in geometry, connecting points, lines, and angles in various contexts.
Midpoint formula: The midpoint formula is a mathematical equation used to find the exact center point between two points in a coordinate plane. It plays a crucial role in geometry, especially in connecting concepts such as distance, line segments, and the equations of shapes like circles. By averaging the x-coordinates and the y-coordinates of two endpoints, the midpoint formula provides essential insights into geometric properties and relationships.
Midpoint of a line segment: The midpoint of a line segment is the point that divides the segment into two equal parts, located exactly halfway between the endpoints. This concept is crucial for understanding how to find the center of a line segment, and it serves as a foundation for further geometric constructions and calculations, such as determining distances and coordinates in a plane.
Points: In geometry, a point is a fundamental unit that represents a precise location in space. It has no dimensions—no length, width, or height—and is typically represented by a dot. Points are essential for defining lines, shapes, and other geometric constructs, serving as the building blocks for more complex structures like angles and polygons.
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