13.1 Distance and midpoint formulas

The distance and midpoint formulas are key tools for working with line segments in a coordinate plane. They help you find lengths between points 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 length of a line segment between two points in a coordinate plane
• Derived from the Pythagorean theorem ($a^2 + b^2 = c^2$) by considering the line segment as the hypotenuse of a right triangle
• Formula: For points $(x_1, y_1)$ and $(x_2, y_2)$, the distance $d$ between them is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• To find the length of a line segment:
  • Identify the coordinates of the endpoints (start and end points)
  • Substitute the $x$ and $y$ 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 $(3, 4)$ and $(7, 1)$
  • $d = \sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Midpoint formula for coordinates

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

Applications of distance and midpoint

• Recognize when to use the distance or midpoint formula 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 $(120, 250)$ and $(400, 150)$ on a map with a scale of 1 unit = 10 miles. Find the distance between the cities.
  • Coordinates: $(120, 250)$ and $(400, 150)$
  • $d = \sqrt{(400 - 120)^2 + (150 - 250)^2} = \sqrt{280^2 + (-100)^2} = \sqrt{78400 + 10000} = \sqrt{88400} \approx 297$ units
  • Scale: 1 unit = 10 miles, so the actual distance is $297 \times 10 \approx 2970$ miles