6.1

Points, Lines and Planes

We study geometry to understand the measurements of shapes and the sizes of objects. The table below describes some fundamental geometrical terms such as point, line, line segment, ray and plane.

Description of objects	Notation
Point has position has no size
Line has an infinite number of points has no width can be straight or curved
Line segment is a part of a straight line between and including two end points has length
Ray is a part of a line with one end point has one end indicated by an arrowhead to show that a ray can be extended in that direction indefinitely
Point of intersection is the point where two lines meet
Plane is a flat surface has no thickness

In this book, a line refers to a straight line and a curve refers to a curved line. When a surface is not flat, it is called a curved surface.

📖 Worked Example 1

The diagram shows four points $P$, $Q$, $R$ and $T$ on a plane. The points $P$, $Q$ and $R$ are on a straight line.

(a) Draw the lines formed by these points. How many lines can be drawn?
(b) Name the line segments that can be formed by the points $P$, $Q$ and $T$.
(c) Name three rays that can be formed by the points $P$, $Q$ and $R$.

SOLUTION

(a) The lines $PR$ (= $PQ$ = $QR$ — these are all the same line through $P$, $Q$, $R$), $PT$, $QT$ and $RT$ are drawn as shown. There are 4 lines.

(b) The line segments formed by $P$, $Q$ and $T$ are $PQ$, $QT$ and $TP$.

(c) Three rays that can be formed by the points $P$, $Q$ and $R$ are $PR$, $RP$ and $QR$ (or many equivalent variants).

📌 NOTE

The rays $PR$ and $RP$ are different — they have different end points and different directions.
The rays $PR$ and $QR$ are different too. Even though they have the same direction, their end points are different.
On the other hand, the lines $PR$, $RP$ and $QR$ all represent the same line. The line passes through all three points and extends in both directions.

✍️ Try It Yourself 1

The diagram shows four points $A$, $B$, $C$ and $D$ on a plane (with no three of them collinear).

(a) How many lines can be drawn? → 检查
💡 思路
每两个点决定一条线。从 4 个点中选 2 个：$\binom{4}{2} = 6$ 条线（$AB, AC, AD, BC, BD, CD$）。
(b) Name the line segments that can be formed by the points $A$, $B$ and $C$. → 检查
💡 思路
三个点两两连接：$AB$, $BC$, $CA$（也可写作 $AC$）。
(c) Name three rays that can be formed by the points $B$, $C$ and $D$ (open question — many answers accepted).

💡 提示
开放题。三个点可以做出 6 条射线：$BC, CB, BD, DB, CD, DC$。任写 3 条即可。

PPractice Exercise 6.1

BASIC MASTERY

Q1. In the diagram, $A$ and $B$ are two points on a plane.

How many lines can be drawn passing through
(a) $A$? → 检查
(b) both $A$ and $B$? → 检查
💡 思路

(a) 过一个点可以画无数条线（方向任意）。
(b) 过两个点恰好一条线——这是几何学的基本公理。
Q2. In the diagram, $C$ and $D$ are two points on a plane. Draw the ray $CD$, where $C$ is the end point.

📐 画一画：从 $C$ 出发，画一条射线穿过 $D$，并在 $D$ 之后用箭头表示"无限延伸"。射线起点 $C$ 不带箭头，终点方向带箭头。

INTERMEDIATE

Q3. In the diagram, $A$, $B$ and $C$ are three points on a plane (no three collinear).

(a) Name the straight lines that can be formed. → 检查
(b) Name the rays with the end point $A$. → 检查
Q4. In the diagram, $A$, $B$ and $C$ are three points on a straight line.

(a) How many different line segments can be formed by these points? → 检查
(b) How many different rays can be formed by these points? → 检查
💡 思路

(a) 线段需要两个端点：$AB, BC, AC$ → 3 条。
(b) 射线需要一个端点 + 一个方向，从同一个点出发的不同方向是不同的射线：
• 从 $A$：$AB$（向 $B$ 方向，包含 $C$），但也可以从 $A$ 向相反方向延伸（这是另一条射线，没经过 $B, C$）。
标准计法：每个端点产生 2 条射线（向两端各一条），共 $3 \times 2 = 6$ 条。
也可只数"经过另外两个点的"：$AB, BA, BC, CB$ 与正向 / 反向射线 ── 仍是 6 条。

ADVANCED

Q5. Look at the picture of a corner of a room (where two walls and the floor meet).

Use the picture above to answer the following questions.
(a) Is it possible for a point to be in two different planes? → 检查
(b) Is it possible for a line to be in two different planes? → 检查
If yes, use the picture to show your answer.

💡 思路

(a) 是。两面墙交线上的每一个点同时在两面墙上（点在两个平面里）。
(b) 是。两面墙的交边（房间的角线）就是一条线，同时在两个平面（两面墙）里。这条线叫做两个平面的"交线"。
Q6. The diagram shows a line segment $ABCD$ in which $AB = CD$.

(a) State the relation between $AC$ and $BD$. → 检查
(b) Explain your answer in (a).

💡 思路

$AC = AB + BC$；$BD = BC + CD$。已知 $AB = CD$，所以 $AC = AB + BC = CD + BC = BC + CD = BD$，即 $AC = BD$。
Q7. (OPEN) The diagram shows a rectangular sheet of paper $ABCD$.

(a) What would you do to the paper so that the point $D$ lies on a different plane from the points $A$, $B$ and $C$?

(b) How do you fold the paper $ABCD$ such that the line segments $AB$ and $DC$ lie on different planes?

💡 思路（开放题）

(a) 在 $D$ 处选一条不经过 $D$ 的折痕（例如沿 $AC$ 对折），把含 $D$ 的那块翻起来。原来共面的 4 个点中 $D$ 就在新的折面上，与 $ABC$ 不共面。
(b) 沿一条平行于 $AB$ 和 $DC$之间的中线（横向折）折起。$AB$ 和 $DC$ 原来都在桌面上，折起一半后变成在两个不同的平面上。