# Line Segment: Definition, Properties & Examples TL;DR A line segment is a part of a straight line bounded by two distinct endpoints, so it has a fixed, measurable length — unlike a line or a ray, which run on forever. This article covers the definition and notation, how a segment differs from a line and a ray, its properties, the distance formula for length, and six worked examples. ## The One Geometric Object Every Shape Is Built From Take any triangle, square, or pentagon apart, and what you are left with is a handful of straight pieces, each pinned down at both ends. Those pieces are line segments, and they are the smallest building block in all of geometry: strip them out, and there is no shape left to talk about. Once you can see why only the bounded piece has a length you can measure, the notation and the distance formula stop being rules to memorise and become things you can reconstruct. ## What Is a Line Segment? A **line segment** is a part of a straight line that is bounded by **two distinct endpoints**, and it contains every point on the line that lies between those two ends. As you trace it, you start at one endpoint and stop at the other; there is no running on forever in either direction. If the endpoints are A and B, the segment is written AB‾, read "line segment AB". The order makes no difference, so AB‾ and BA‾ name the same set of points. The **length** of the segment is written AB without the bar: AB‾ is the geometric object, while AB is the number measuring how long it is. ## How a Line Segment Differs From a Line and a Ray A reader audit kept surfacing one question above all others here, so it is worth answering head-on. **What is the difference between a line, a ray, and a line segment?** All three live on the same straight path; the difference is only where they start and stop. | Object | Notation | Endpoints | Length | |----------------|-------------------------------------|------------------------------------|----------------| | **Line** | AB↔ | None, runs both ways forever | Infinite | | **Ray** | AB→ | One, starts at A and runs through B forever | Infinite | | **Line segment**| AB‾ | Two, fixed at A and B | Finite, measurable | The fastest way to read the notation: count the arrowheads. Two arrowheads (AB↔) means a line, one arrowhead (AB→) means a ray, and a plain bar (AB‾) means a segment. Only the segment can be laid against a ruler and given a length. ## Properties of a Line Segment Everything special about a line segment comes from the fact that it is closed off at both ends. The properties below are just that one idea, seen from different angles. - **Fixed, measurable length.** The length of AB‾ stays the same no matter which way you measure it or where you slide or rotate it. A line and a ray have no finite length to speak of. - **Congruence.** Two segments are **congruent** (AB‾≅CD‾) when they have equal lengths. In a square, all four sides are congruent segments; in an equilateral triangle, all three are. - **Parallel segments.** Two segments are **parallel** (∥) when the gap between them stays constant, so extended they would never meet. Opposite sides of a rectangle or parallelogram are parallel. - **Perpendicular segments.** Two segments are **perpendicular** (⊥) when they meet at a right angle (90°). Adjacent sides of a square meet this way. (For the line that cuts a segment in half at a right angle.) - **Intersecting segments.** Two segments **intersect** when they share at least one point. The two diagonals of a rectangle cross at a single interior point. - **Midpoint.** Every segment has exactly one **midpoint** — the point that splits it into two congruent halves. ## The Length of a Line Segment: the Distance Formula When you can lay a ruler against a drawn segment, you simply read the length off. When the endpoints are given as coordinates (x1,y1) and (x2,y2) on the plane, you compute the length with the **distance formula**: d=(x2−x1)²+(y2−y1)².\ Here x2−x1 is the horizontal gap between the endpoints and y2−y1 is the vertical gap. The formula is the Pythagorean theorem in disguise: the segment is the hypotenuse of a right triangle whose two legs are those horizontal and vertical gaps, so d²=(horizontal)²+(vertical)², and taking the square root gives d. ## Examples of the Line Segment ### **Example 1:** Identify whether PQ‾, PQ→, and PQ↔ each have a measurable length. Only PQ‾ does. The bar means a segment with two endpoints, so it has a finite length. The single arrow PQ→ is a ray and the double arrow PQ↔ is a line; both run on forever and have no finite length. Final answer: only PQ‾ has a measurable length. ### **Example 2:** Find the length of the segment between P(−3,4) and Q(5,−2). A common first move is to write d=(5−(−3))²+(−2−4)²=8²+(−6)²=64+36=100=10. Final answer: d=10 units. ### **Example 3:** Find the length of the segment between A(1,2) and B(4,6). d=(4−1)²+(6−2)²=9+16=25=5. Final answer: d=5 units. ### **Example 4:** The midpoint of AB‾ is M(3,5) and one endpoint is A(1,2). Find the other endpoint B. The midpoint is the average of the endpoints, so B=(2⋅3−1,2⋅5−2)=(5,8). Final answer: B=(5,8). ### **Example 5:** Two segments measure CD‾=7 cm and EF‾=7 cm. Are they congruent, and does congruence mean they sit in the same place? They are congruent, written CD‾≅EF‾, because they have equal lengths. Congruence is about length only, so the two segments can point in different directions or sit far apart and still be congruent. Final answer: yes, congruent; congruence fixes length, not position. ### **Example 6:** A segment has length 13 units. One endpoint is A(2,3) and the other has the form (x,8). Find x. Apply the distance formula and square both sides: 13=(x−2)²+(8−3)²;⇒;169=(x−2)²+25;⇒;(x−2)²=144. So x−2=±12, giving x=14 or x=−10. Final answer: x=14 or x=−10. ## Where Line Segments Show Up A line segment is the most basic object in geometry, which is exactly why it turns up wherever a fixed distance between two points matters. The reach goes well past the textbook. - **Engineering and CAD drawings.** Every edge of every part on a computer-aided design is a line segment carrying a precise length tolerance. - **Computer graphics.** A "line" drawn on a screen is really a segment, and drawing apps run the Bresenham algorithm to decide which pixels to colour between its two endpoints. - **GPS routing.** Mapping apps approximate roads as chains of segments; the trip distance is the sum of all the segment lengths. - **Land surveying.** Surveyors record plot boundaries as segments and add their lengths to find a property's perimeter. ## Where Students Trip Up on Line Segments ### **Mistake 1: Treating a line or a ray as if it had a finite length** **Where it slips in:** A problem asks for the length of AB↔, and a student computes a number anyway. **The correct way:** Read the notation first. Only AB‾ (the bar) has a finite length. AB→ and AB↔ run on forever. ### **Mistake 2: Dropping a negative sign in the distance formula** **Where it slips in:** Computing (x2−x1) when x1 is negative, writing (5−3) instead of (5−(−3)). **The correct way:** Write (x2−x1) with the actual values bracketed, then simplify. ### **Mistake 3: Forgetting the square root at the end** **Where it slips in:** Reporting 169 as the length. **The correct way:** Take the positive square root: d=√169=13. A length is always positive. ## Key Takeaways - A **line segment** is a part of a straight line bounded by two distinct endpoints, with a fixed, measurable length. - It differs from a line (runs both ways forever) and a ray (runs one way forever); only the segment has a finite length. - Segments can be congruent, parallel, perpendicular, or intersecting, and each has exactly one midpoint. - The length between coordinates (x1,y1) and (x2,y2) is given by the distance formula d=(x2−x1)²+(y2−y1)². - The most common slip is dropping a negative sign in the subtraction; write the brackets before you compute.