Coordinate Pairs and Cartesian Point Relations

Page 1 Data Records:
- Item 1: Numerical pair association involving coordinates $(2, 1)$ and value $5.3$ (or point $(5, 3)$ ).
- Item 31: Point located at coordinates $(2, 5)$ .
- Item 3: Vector or coordinate tuple represented as $(1, -45, -2)$ .
- Item 4: Pair of coordinate points $(-3, 1)$ and $(-7, 4)$ .
- Item 8.2: Expression or coordinate pair represented by $(8, -1)$ .
- Item 6: Scalar value $-58$ paired with point $(-4, 2)$ .
- Item (E) 10-11: Entry associated with point $(4, -7)$ .
- Item 4 (variant): Coordinate sequence connecting point $(1, -1)$ to incomplete coordinates $(-2, -)$ .
- Item 19: Four-dimensional coordinate tuple or set of values $(-4, -1, -2, 0)$ .
- Item 10: Point $(-3, -3)$ associated with scalar value $1.0$ .
- Item 12.5: Point coordinates $(1, 1)$ .
- Item (18): Point $(10, 0)$ linked with scalar value $-27$ .

Distance Formula:
- The standard Euclidean distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ in two-dimensional space is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
- Applying this to the given point pair $(-3, 1)$ and $(-7, 4)$ : $d = \sqrt{(-7 - (-3))^2 + (4 - 1)^2}$ $d = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$
Midpoint Formula:
- The midpoint $M$ of a segment connecting $(x_1, y_1)$ and $(x_2, y_2)$ is defined as: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
- Applying this to the points $(-3, 1)$ and $(-7, 4)$ : $M = \left(\frac{-3 + (-7)}{2}, \frac{1 + 4}{2}\right) = \left(-5, \frac{5}{2}\right)$
Slope Formula:
- The slope $m$ of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1}$
- Applying this to the points $(-3, 1)$ and $(-7, 4)$ : $m = \frac{4 - 1}{-7 - (-3)} = \frac{3}{-4} = -\frac{3}{4}$