Grade 4 · 4.G.A.1

Points, lines, rays and segments

A star map joins stars with straight marks, and a telescope’s line of sight goes on into space. How do you say exactly where a straight path starts, where it stops, and whether it ever stops at all?

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Why this exists

Nevena, Bindi and Jedda run an astronomy club. On their star map each star is a tiny dot, and the shapes of the constellations are drawn by joining some of the dots with straight marks. When Bindi aims the club’s telescope, she looks from where she stands, past one star, and on out into space.

To talk about these drawings, the club needs exact words. “The straight bit” is not enough. One straight mark stops at two stars. Bindi’s line of sight starts at her eye and never stops. Another straight path might go on forever in both directions. These are three different things, and they get three different names. The place where a star sits gets a fourth name.

Those four words are point, line, ray and segment. Every shape in geometry is built from them. A triangle’s sides are segments. An angle is made of two rays. A square has corners that are points.

The lesson takes two days. Part 1 is the four objects: what each one is, how it is drawn, and how it is named with letters and symbols. Part 2 is using the names: counting the segments and rays in a figure, and finding them in drawings.

Definition

Part 1: the four objects.

A point is an exact location. It has no size at all; the dot you draw only shows where it is. A point is named with one capital letter, such as point A.

A line is a straight path that goes on forever in both directions. It has no ends. It is drawn with an arrowhead at each end to show that it keeps going. A line is named by any two points on it, with a two-headed arrow over the letters: \overleftrightarrow{BC}, read “line BC”. A line can also be named by one lowercase letter, such as line m.

A ray is a straight path that starts at one point, called its endpoint, and goes on forever in one direction. It is drawn with a dot at the endpoint and an arrowhead at the other end. A ray is named by its endpoint first, then any other point on it, with a one-headed arrow over the letters pointing to the right: \overrightarrow{DE}, read “ray DE”.

A segment is a straight path that stops at two endpoints. It is drawn with a dot at each end and no arrowheads. A segment is named by its two endpoints, with a bar over the letters: \overline{FG}, read “segment FG”.

Figure 1The four objects. A dot means “it stops here”; an arrowhead means “it keeps going”.

Object	Ends	Drawn with	Name
point	none	one dot	A
line	0	two arrowheads	\overleftrightarrow{BC}
ray	1	dot, arrowhead	\overrightarrow{DE}
segment	2	two dots	\overline{FG}

The arrow over a ray’s name always points to the right, whichever way the ray points on the page. The order of the letters is what says where it starts.

Part 2: counting them. When a figure has named points, a segment “in the figure” is one whose two endpoints are named points and that lies along something drawn. A ray in the figure starts at a named point and runs along something drawn.

Intuition

Think of an arrowhead as the word “and on and on”, and a dot at the end of a stroke as the word “stop”. Read a drawing that way and the four objects sort themselves out.

On the club’s star map, every star is a point. The marks that join stars stop at a star at each end, so they are segments. Bindi’s line of sight starts at her telescope and goes on past the stars into space, so it is a ray. A line is harder to find in real life, because almost nothing goes on forever both ways. You can picture one by taking a ray and letting it run backward past its endpoint too.

The three straight objects nest inside one another. Pick two points A and B. The line \overleftrightarrow{AB} is the longest: all of it. The ray \overrightarrow{AB} is the part of that line that starts at A and heads toward B and beyond. The segment \overline{AB} is the part between A and B, and nothing more. A segment is the only one of the three you can lay a ruler along from end to end.

Results

One line through two points. Through any two different points there is exactly one line. That is why two letters are enough to name a line.

Which names can be turned around. For two points A and B:

\overline{AB} = \overline{BA} \qquad \overleftrightarrow{AB} = \overleftrightarrow{BA}

but \overrightarrow{AB} and \overrightarrow{BA} are different rays. One starts at A; the other starts at B and points the other way.

Many names for one ray. If B and C are both on the same side of A along one line, then \overrightarrow{AB} and \overrightarrow{AC} are the same ray. The first letter has to be the endpoint; the second can be any other point on the ray.

Counting. Every point on a line is the endpoint of exactly two rays along that line, one going each way. When some points are named and every pair of them is joined by a segment, count the pairs: the first point pairs with each point after it, the second with each point after it, and so on. For four points that is 3 + 2 + 1 = 6 segments.

Why it is true

One line through two points. Lay a ruler so its edge touches two dots. There is only one way to do it: if you tip the ruler even a little, its edge misses one of the dots. Geometry takes this as one of its starting facts.

Why a ray’s name has an order. The two letters of a ray do different jobs. The first says where it starts; the second only says which way it goes. Swap them and you name a ray that starts somewhere else and points the other way. The two letters of a segment do the same job (each one is an end), so swapping them changes nothing. A line has no ends, and its two letters just say which line it is.

Why counting pairs works. Each segment has two endpoints, so each segment is one pair of points. Listing the pairs in order, first everything that starts at the first point, then everything new that starts at the second point, and so on, reaches every pair once and never twice.

Worked examples

Examples 1 to 4 go with Part 1, and Examples 5 to 7 with Part 2.

Example 1

Reading a star map

Nevena draws a W of five stars on the club’s map (Figure 2). How many points are in the drawing? How many segments? Name each segment, and say why they are segments and not lines or rays.

Figure 2Nevena’s drawing of a W of five stars.

Solution

Each star is a point, and the stars are labeled Q, R, S, T and U. That is 5 points.

Count the straight marks. One joins Q to R, one R to S, one S to T and one T to U. Five stars in a chain need 5 - 1 = 4 joins, so there are 4 segments:

\overline{QR} \qquad \overline{RS} \qquad \overline{ST} \qquad \overline{TU}

Each mark stops at a star at both ends, and there is no arrowhead anywhere. Two endpoints make a segment. Notice that \overline{QS} is not in the drawing: Q and S are both points on the map, but no mark joins them.

Example 2

Naming a ray the right way round

Bindi’s line of sight starts at point T and passes through two stars, U and V (Figure 3). Which of these name the ray drawn: \overrightarrow{TU}, \overrightarrow{UT}, \overrightarrow{TV}, \overrightarrow{UV}?

Figure 3Bindi’s line of sight starts at T and passes through U and V.

Solution

The ray’s endpoint is T: that is where the dot is, and the arrowhead is at the other end. So its name must start with T.

\overrightarrow{TU} starts at T and goes through U. That is the ray drawn.

\overrightarrow{TV} starts at T and goes through V. Since V is on the same ray, this is the same ray again, with a different second letter.

\overrightarrow{UT} starts at U and heads toward T, then on past T. It points the opposite way. It is not the ray drawn.

\overrightarrow{UV} points the right way, but it starts at U. It leaves out the piece from T to U, so it is a smaller ray sitting inside the one drawn, not the same one.

The ray drawn is \overrightarrow{TU}, also called \overrightarrow{TV}.

Example 3

Same figure or different?

Use Figure 4, a line with points A, B and C on it in that order. For each pair, say whether the two names mean the same figure.

Figure 4One line with three named points on it.

\overline{AB} and \overline{BA}
\overrightarrow{AB} and \overrightarrow{BA}
\overleftrightarrow{AB} and \overleftrightarrow{CA}
\overrightarrow{AB} and \overrightarrow{AC}
\overline{AB} and \overline{AC}

Solution

Same. Both are the straight path between A and B. A segment’s two letters are its two ends, and the order does not matter.
Different. \overrightarrow{AB} starts at A and goes right. \overrightarrow{BA} starts at B and goes left. They share only the piece between A and B.
Same. A, B and C all sit on the one line drawn, and any two of them name it.
Same. Both start at A. Both head right, since B and C are both to the right of A.
Different. Both start at A, but \overline{AB} stops at B and \overline{AC} goes on to C. They have different endpoints, so \overline{AC} is longer.

Example 4

Drawing from a description

Draw \overleftrightarrow{MN}. Mark a point O that is not on it. Then draw \overrightarrow{OM} and \overline{ON}.

Solution

Read each symbol for what it asks for.

\overleftrightarrow{MN} has the two-headed arrow, so it is a line. Mark two dots, M and N, draw a straight stroke through them, and put an arrowhead at each end.

Point O is a single dot, anywhere off that line. Figure 5 puts it above.

\overrightarrow{OM} is a ray whose endpoint is O, the first letter. Put the stroke’s start at O, run it straight through M, and keep going past M. It crosses the line at M and ends in an arrowhead on the far side. Stopping at M would draw \overline{OM} instead.

\overline{ON} is a segment, so draw a straight stroke from O to N that stops at both.

Figure 5Line MN, ray OM and segment ON, drawn from the description.

Check. The drawing has 3 named points, 1 line, 1 ray and 1 segment, as the description asked.

Example 5

Counting segments on a subway track

Jedda draws a straight subway track with four stations, F, G, H and J, in a row (Figure 6). How many segments have two of these stations as endpoints? List them.

Figure 6Jedda’s subway track with four stations.

Solution

Every pair of stations is the two ends of one segment along the track, so count the pairs in an order that cannot miss one or repeat one.

Start at F. It pairs with each station after it: \overline{FG}, \overline{FH}, \overline{FJ}. That is 3.

Move to G. Its pair with F is already listed, so take only the stations after it: \overline{GH}, \overline{GJ}. That is 2.

Move to H. Only J is left: \overline{HJ}. That is 1.

J has no stations after it, so nothing new starts there. In all:

3 + 2 + 1 = 6 \text{ segments}

The track itself is one of them, \overline{FJ}. The other five are pieces of it. A segment between two stops is still a segment when other stops sit in the middle of it.

Example 6

Counting rays on a line

Use Figure 4 again: a line with points A, B and C. How many rays start at one of these three points and run along the line? How many of them can be named with two of the letters?

Solution

Stand at any one point on a line. The line runs two ways from there, so the point is the endpoint of exactly 2 rays along it. Three points give

3 \times 2 = 6 \text{ rays}

Now try to name each one with two of the letters. The second letter must be a point on the ray.

From A: going right, the ray passes B and C, so it is \overrightarrow{AB} (also called \overrightarrow{AC}). Going left, it passes no named point, so it has no two-letter name in this figure.

From B: going left it is \overrightarrow{BA}, and going right it is \overrightarrow{BC}.

From C: going left it is \overrightarrow{CB} (also called \overrightarrow{CA}). Going right, no named point is on it.

So there are 6 rays, and 4 of them can be named with two letters. The six two-letter names \overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{BA}, \overrightarrow{BC}, \overrightarrow{CA}, \overrightarrow{CB} name only those 4 rays, because \overrightarrow{AB} and \overrightarrow{AC} are one ray, and so are \overrightarrow{CA} and \overrightarrow{CB}. Counting names is not the same as counting rays.

Example 7

Joining every pair of stars

The club wants to join five stars, K, L, M, N and P, so that every pair has a segment between them (Figure 7). No three of the stars lie on one line. How many segments will they draw?

Figure 7Five stars with every pair joined. Five sides and five segments across the inside.

Solution

These stars are not in a row, but the counting works the same way as on the subway track: every pair of points gets one segment.

K pairs with the four stars after it: \overline{KL}, \overline{KM}, \overline{KN}, \overline{KP}. Then L pairs with the three after it: \overline{LM}, \overline{LN}, \overline{LP}. Then M with two: \overline{MN}, \overline{MP}. Then N with one: \overline{NP}.

4 + 3 + 2 + 1 = 10 \text{ segments}

Check. Count the picture another way. The five-sided shape has 5 sides, and 5 more segments cross its inside, one from each corner to each corner that is not next to it. That is 5 + 5 = 10, the same answer.

Boundaries

A drawing is a picture of the idea, not the idea. A dot on paper has a size, and a point has none. A drawn line stops at the edge of the page; the arrowheads are how the drawing says the line does not. So read the marks, not the length of the stroke: a short stroke with two arrowheads is a line, and a long stroke with a dot at each end is a segment.

Straight only. All four objects are straight. A curved path between two stars is not a segment, and a path that bends at a corner is two segments, one on each side of the corner.

A ray’s second letter must be on the ray. \overrightarrow{AB} only makes sense if B is on the ray. A point on the line but behind the endpoint belongs to a different ray, the one pointing the other way. Two points that are the same point cannot name a line, a ray or a segment at all.

What gets counted depends on what is named. Example 6 found 6 rays but could name only 4. A figure has more segments and rays than its letters can name, so a counting question always says which points count.

Everyday words do not match. A subway “line” bends and has two end stations, so in geometry it is not a line at all; a straight stretch of it between two stations is a segment.

Where this goes next

Two rays that share an endpoint make an angle, and the endpoint is the angle’s vertex. Naming an angle uses the same letters as naming a ray, with the vertex in the middle: angles and how to name them picks up from here, and the next lessons measure how far one ray turns from the other.

Two lines in a flat surface either cross or never meet. When they cross at a square corner they are perpendicular; when they never meet, however far they go, they are parallel. Both ideas need lines that go on forever, which is why a line was defined that way.

Much later, geometry looks again at the starting words point and line and asks why they are left without a definition; that is points, lines and planes. A segment’s length, and how lengths along one line add up, is in segments, rays and length.

Exercises

Part 1: which object is it?

A dot on Nevena’s star map shows where one star is. Is it a point, a line, a ray or a segment?
A thread is pulled tight between two pins. Which of the four objects is it a model of?
A drawing shows a straight stroke that starts at a dot and ends in an arrowhead. What is it?
How many endpoints does a line have? A ray? A segment?

Part 1: names and symbols

Write each with symbols: line GH, ray HG, segment GH.
Bindi drew a ray that starts at point W and passes through point X. She labeled it \overrightarrow{XW}. Is the label right? If not, fix it.
True or false: \overline{XY} and \overline{YX} name the same segment; \overrightarrow{XY} and \overrightarrow{YX} name the same ray; \overleftrightarrow{XY} and \overleftrightarrow{YX} name the same line.
Points D, E and F lie on a line in that order. Do \overrightarrow{DE} and \overrightarrow{DF} name the same ray? Do \overrightarrow{ED} and \overrightarrow{EF}?

Part 1: reading and drawing a figure

Use Figure 8 for Exercises 9 to 13.

Figure 8A line, a ray, a segment and a point on its own, for Exercises 9 to 13, 17 and 18.

Name the line in three different ways.
Name the ray that rises from the line.
Name the segment that does not lie along the line or the ray.
Which named point is not on any line, ray or segment?
Is \overrightarrow{SQ} drawn in Figure 8? Explain.
Draw point K, then \overrightarrow{KL}, then \overline{LM} with M not on \overrightarrow{KL}. Describe what your drawing shows.

Part 2: counting segments and rays

Use Figure 9, the Big Dipper, for Exercises 15 and 16.

Figure 9The Big Dipper as a star map draws it, for Exercises 15 and 16.

How many points and how many segments are in Figure 9?
Which star in Figure 9 is an endpoint of three segments? Name the three segments.
In Figure 8, how many segments have two of the points P, Q and R as endpoints? Name them.
In Figure 8, how many rays start at Q and run along something drawn? Name them.
A subway line has 7 stations in a straight row. How many segments have two of the stations as endpoints?
Six stars, no three in a line, are joined so that every pair has a segment between them. How many segments is that?
Four points are marked on a line. How many rays start at one of the four points and run along the line?

Reasoning

You can measure a segment with a ruler. Explain why you cannot measure a line or a ray the same way.
Nevena says \overrightarrow{PQ} and \overrightarrow{PR} must be different rays, because the letters are different. Explain when she is right and when she is wrong.
Explain why the order of the letters matters in the name of a ray but not in the name of a segment.
Jedda says that adding an eighth station to the end of the 7-station subway line in Exercise 19 adds exactly 7 new segments. Is she right? Explain.

Answers

A point. The dot marks one exact location; how big the dot is drawn does not matter.
A segment. The thread is straight and stops at the two pins, which are its two endpoints.
A ray. The dot is its one endpoint, and the arrowhead says it goes on forever that way.
A line has 0 endpoints, a ray has 1 and a segment has 2.
\overleftrightarrow{GH}, \overrightarrow{HG} and \overline{GH}. The ray starts at H, so H comes first.
No. A ray is named endpoint first, so it is \overrightarrow{WX}. The label \overrightarrow{XW} names a ray that starts at X and points the other way.
True, false, true. Only the ray’s letters have an order: \overrightarrow{XY} starts at X and \overrightarrow{YX} starts at Y.
Yes, then no. \overrightarrow{DE} and \overrightarrow{DF} both start at D and head the same way, since E and F are on the same side of D. \overrightarrow{ED} and \overrightarrow{EF} both start at E but point in opposite directions.
Any three of \overleftrightarrow{PQ}, \overleftrightarrow{QP}, \overleftrightarrow{PR}, \overleftrightarrow{RP}, \overleftrightarrow{QR}, \overleftrightarrow{RQ}. Any two of the three points on it name it.
\overrightarrow{QS}. Its endpoint is Q, so Q comes first.
\overline{ST}, which can also be written \overline{TS}.
U.
No. \overrightarrow{SQ} would start at S, go down through Q and keep going below the line. The figure has nothing below the line there: the ray drawn starts at Q and goes up, so it is \overrightarrow{QS}.
A dot K; a straight stroke from K through a dot L that keeps going to an arrowhead past L; and a dot M off that stroke, with a straight stroke from L to M that stops at both, with no arrowheads.
7 points and 7 segments: 4 around the bowl (\overline{AB}, \overline{BC}, \overline{CD}, \overline{DA}) and 3 in the handle (\overline{DE}, \overline{EF}, \overline{FG}).
D. Its three segments are \overline{DA}, \overline{DC} and \overline{DE}.
3 segments: \overline{PQ}, \overline{QR} and \overline{PR}. P pairs with 2 points after it and Q with 1, and 2 + 1 = 3.
3 rays: \overrightarrow{QP} and \overrightarrow{QR} along the line, one each way, and \overrightarrow{QS} rising from it.
21 segments. The first station pairs with the 6 after it, the next with 5, and so on: 6 + 5 + 4 + 3 + 2 + 1 = 21.
15 segments. 5 + 4 + 3 + 2 + 1 = 15, the same pair counting as on a track.
8 rays. Each point is the endpoint of 2 rays along the line, and 4 \times 2 = 8.
Measuring a length means finding where something starts and where it stops. A segment has two endpoints, so a ruler can run from one to the other. A ray has only one endpoint and a line has none: they go on forever, so there is no other end to read off the ruler.
Both rays start at P. She is wrong when Q and R are on the same side of P along one line: then both names start at P and head the same way, so they are the same ray. She is right when Q and R are on opposite sides of P, or are not on one line with P at all.
A ray’s first letter is its endpoint and its second only shows the direction, so the two letters do different jobs and swapping them names a ray that starts somewhere else. A segment’s two letters are both endpoints and do the same job, so swapping them describes the same segment.
Yes. Every old segment is still there, and the only new segments are the ones with the new station as an endpoint. The new station pairs once with each of the 7 old stations, so it adds 7. Check: 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28, and 28 - 21 = 7.