Points, Lines, Segments, and Planes

Geometry begins with a few simple, undefined terms that build the foundation for everything else. Understanding these basic objects and how to measure them is essential for studying shapes, angles, and space.

The Building Blocks of Geometry

• Point: A specific location in space. It has no size, length, width, or depth. It is usually represented by a dot and named with a capital letter (e.g., Point $A$).
• Line: A straight, continuous arrangement of infinitely many points that extends forever in two opposite directions. It has length but no width or thickness.
• Line Segment: A portion of a line consisting of two endpoints and all the points between them. Unlike a line, a segment has a measurable length.
• Plane: A flat, two-dimensional surface that extends infinitely in all directions. You can think of it as a never-ending sheet of paper.

Segment Addition Postulate

The Segment Addition Postulate is a simple but powerful rule. It states that if three points $A$, $B$, and $C$ are collinear (on the same line) and $B$ is between $A$ and $C$, then the sum of the lengths of the two smaller segments equals the length of the entire segment:

$AB + BC = AC$

The Midpoint Formula

The midpoint is the exact middle point of a line segment, dividing it into two equal parts. On a coordinate plane, you can find the midpoint $M$ of a segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ by averaging their $x$ and $y$ coordinates:

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

Example: Find the midpoint of $A(3, -2)$ and $B(7, 4)$.

1. Add the $x$-coordinates and divide by 2: $\frac{3 + 7}{2} = \frac{10}{2} = 5$
2. Add the $y$-coordinates and divide by 2: $\frac{-2 + 4}{2} = \frac{2}{2} = 1$

The midpoint is $(5, 1)$.

The Distance Formula

To find the exact length of a line segment on a coordinate plane, we use the distance formula (which is derived from the Pythagorean theorem). The distance $d$ between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is:

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

Example: Find the distance between $P(1, 3)$ and $Q(4, 7)$.

1. Subtract the $x$-coordinates and square the result: $(4 - 1)^2 = 3^2 = 9$
2. Subtract the $y$-coordinates and square the result: $(7 - 3)^2 = 4^2 = 16$
3. Add the values and take the square root: $d = \sqrt{9 + 16} = \sqrt{25} = 5$

The distance between the two points is $5$ units.