Coordinate Geometry, Distance, and Midpoint Formulas

Number Line Representation and Horizontal Axis

  • Structure of the horizontal number line:

    • The center point or origin is marked at 00

    • Positive values extend along the right side of the horizontal line

    • Negative values extend along the left side of the horizontal line

    • Exact positioning along the line requires measuring distances relative to the origin at 00

Coordinate Signs and Quadrants

  • Coordinate pairs and sign designations:

    • A standard coordinate pair is composed of a first coordinate (xx-axis value) and a second coordinate (yy-axis value)

    • The sign of the first coordinate indicates horizontal direction (positive or negative)

    • The sign of the second coordinate indicates vertical direction (positive or negative)

    • When both the first coordinate and the second coordinate are negative, the point lies in the negative-negative coordinate region with signs (−,−)(-, -)

Symmetry Around the Origin

  • Properties of origin symmetry:

    • Symmetry around the origin (0,0)(0, 0) differs from line symmetry around an exact horizontal or vertical line

    • Reflecting a point (x,y)(x, y) symmetrically around the origin yields the point (−x,−y)(-x, -y)

Distance Formula Construction

  • Step-by-step derivation of the distance formula between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2)

    • Step 1: Compute the difference between the xx-coordinates:     x2−x1x_2 - x_1

    • Step 2: Square the xx-coordinate difference:     (x2−x1)2(x_2 - x_1)^2

    • Step 3: Compute the difference between the yy-coordinates:     y2−y1y_2 - y_1

    • Step 4: Square the yy-coordinate difference:     (y2−y1)2(y_2 - y_1)^2

    • Step 5: Add the two squared differences together:     (x2−x1)2+(y2−y1)2(x_2 - x_1)^2 + (y_2 - y_1)^2

    • Step 6: Take the principal square root of the combined sum to find the total distance dd     d = \n\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Midpoint Formula Justification

  • Algebraic logic for division by 22

    • Adding two identical variables together results in two times the quantity:     x+x=2xx + x = 2x

    • Because adding equal segments doubles the value, finding the exact central point or average position requires dividing the sum by 22

    • The midpoint coordinates MM between (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are:     M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)