Having two coordinate points

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having two coordinate points

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LectureNotes said having two coordinate points

Answer:
When discussing coordinate points in a lecture or educational context, it’s essential to understand the basics of coordinate geometry. Coordinate geometry, or analytic geometry, involves the study of geometry using a coordinate system, typically the Cartesian coordinate system.

What are Coordinates?

Coordinates are a set of values that show an exact position. On a two-dimensional plane, which is the most common context for introductory discussions, an ordered pair (x, y) represents a point. The x coordinate tells you how far along the horizontal axis (often called the x-axis) the point is, and the y coordinate tells you how far along the vertical axis (the y-axis) the point is.

Using Two Coordinate Points

When you have two coordinate points, you can perform a variety of geometric operations. Common uses include:

  1. Calculating Distance Between Two Points:

    • The distance between two points (x_1, y_1) and (x_2, y_2) can be found using the distance formula:

      d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

      This formula is derived from the Pythagorean theorem.

  2. Finding the Midpoint:

    • The midpoint of the line segment connecting two points (x_1, y_1) and (x_2, y_2) is given by:

      \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

    • This is simply the average of the x-coordinates and the y-coordinates of the two points.

  3. Determining the Slope of the Line:

    • The slope m of the line passing through the points (x_1, y_1) and (x_2, y_2) is:

      m = \frac{y_2 - y_1}{x_2 - x_1}

    • This formula represents the rate of change of y with respect to x and is fundamental in understanding linear relationships.

  4. Equation of a Line:

    • Once you have the slope, the equation of the line passing through (x_1, y_1) and (x_2, y_2) can be written in the slope-intercept form y = mx + b or point-slope form:

      y - y_1 = m(x - x_1)

Example:

Let’s apply these concepts with specific points. Assume we have two points A(1, 2) and B(4, 6).

  1. Calculating Distance:

    • Plug in the coordinates into the distance formula:

      d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

  2. Finding Midpoint:

    • Compute the midpoint:

      \left( \frac{1 + 4}{2}, \frac{2 + 6}{2} \right) = \left( \frac{5}{2}, \frac{8}{2} \right) = \left( 2.5, 4 \right)

  3. Determining the Slope:

    • Calculate the slope:

      m = \frac{6 - 2}{4 - 1} = \frac{4}{3}

  4. Equation of the Line:

    • Using the slope and the point (1, 2):

      y - 2 = \frac{4}{3}(x - 1) \implies y = \frac{4}{3}x - \frac{4}{3} + 2 \implies y = \frac{4}{3}x + \frac{2}{3}

Final Answer:
Understanding the uses and calculations involving two coordinate points in the Cartesian plane is fundamental in geometry and is crucial for progressing to more advanced topics in mathematics and related fields.