Cartesian Coordinate System Distance Between Two Points

Overview

Engineering reference data for Cartesian Coordinate System Distance Between Two Points in mathematics.

Key Formulas

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Roots of ax² + bx + c = 0.

Pythagorean Theorem

$c^2 = a^2 + b^2$

Right triangle relationship.

Circle Area

$A = \pi r^2$

Area of a circle.

Logarithm

$\log_b(x) = \frac{\ln(x)}{\ln(b)}$

Change of base formula.

Variables

Intermediate Position Calculation

The point P that lies on the line segment connecting points P₁ and P₂ can be calculated using a ratio. Let r₁ be the ratio of the distance from P₁ to P, and r₂ be the ratio of the distance from P₂ to P, relative to the total distance between P₁ and P₂.

The coordinates of the intermediate point P are given by: $x = \frac{r_1 x_1 + r_2 x_2}{r_1 + r_2}$ $y = \frac{r_1 y_1 + r_2 y_2}{r_1 + r_2}$

Midpoint Formula

For the exact midpoint between two points, the ratios are equal: $r_1 = r_2 = 1$. This simplifies the intermediate position formulas to the standard midpoint formulas: $x = \frac{x_1 + x_2}{2}$ $y = \frac{y_1 + y_2}{2}$

Using the example points P₁(4, 5) and P₂(7, 9):

• Midpoint x-coordinate: $x = \frac{4 + 7}{2} = 5.5$
• Midpoint y-coordinate: $y = \frac{5 + 9}{2} = 7$ The midpoint is located at (5.5, 7).

LaTeX Formulas and Definitions

Distance Formula

The distance between two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ in Cartesian coordinates is:

$$d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Intermediate Position Formulas

For an intermediate point $P$ dividing the segment between $P_1$ and $P_2$ with ratios $r_1$ and $r_2$, the coordinates are:

$$x = \frac{r_1 x_1 + r_2 x_2}{r_1 + r_2}$$ $$y = \frac{r_1 y_1 + r_2 y_2}{r_1 + r_2}$$

where:

• $r_1$: ratio of the distance from $P_1$ to $P$ relative to the distance from $P_1$ to $P_2$.
• $r_2$: ratio of the distance from $P_2$ to $P$ relative to the distance from $P_1$ to $P_2$.

Midpoint Simplification

When $r_1 = r_2 = 1$ (midpoint), the formulas reduce to:

$$x = \frac{x_1 + x_2}{2}$$ $$y = \frac{y_1 + y_2}{2}$$

Definitions

In a rectangular Cartesian coordinate system, the coordinate axes (x-axis and y-axis) are perpendicular to each other, and the same unit length is consistently used on both axes.

Formulas

Distance Between Two Points:

$$d(P_1,P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Weighted Intermediate Position:

$$x = \frac{r_1 x_1 + r_2 x_2}{r_1 + r_2}$$ $$y = \frac{r_1 y_1 + r_2 y_2}{r_1 + r_2}$$

Midpoint (where $r_1 = r_2 = 1$):

$$x = \frac{x_1 + x_2}{2}$$ $$y = \frac{y_1 + y_2}{2}$$

Practical Examples

Distance Calculation: For points $P_1(4, 5)$ and $P_2(7, 9)$:

$$d(P_1,P_2) = \sqrt{(7 - 4)^2 + (9 - 5)^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$$

Midpoint Calculation: For the same points:

$$x = \frac{4 + 7}{2} = 5.5$$ $$y = \frac{5 + 9}{2} = 7$$

References

• Original Source