Free Lines Rays and Line Segments Worksheet | 10+ Pages - Free Printable

Let's analyze the given complex figure step by step and answer each question carefully.

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Given Figure Analysis:

We have a geometric figure with points labeled:
- A, B, C, D, E, N, O
- Lines and rays extending in various directions
- Some lines intersect at point O, which seems to be the central intersection point.

Let’s go through each part:

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i) Name the two intersecting lines?

Answer:
The two lines that intersect at point O are:

- Line AB (horizontal line passing through A, O, B)
- Line CE (diagonal line passing through C, O, E)

These two lines cross at point O, so they are intersecting lines.

✔ Answer: AB and CE

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ii) How many line segments in the above figure?

A line segment has two endpoints.

Let’s list all distinct line segments visible in the figure:

1. AO (from A to O)
2. OB (from O to B)
3. OC (from O to C)
4. OD (from O to D)
5. OE (from O to E)
6. ON (from O to N)
7. DB (from D to B) — this is a vertical segment
8. CD (from C to D) — horizontal segment between C and D

Wait — let’s check if CD is a separate segment or part of a longer line.

But note:
- C and D are on a horizontal line, but only CD is shown as a segment.
- Also, D to B is a vertical segment.

But also consider:
- Is CO and OD part of one line? Yes, but CO and OD are segments.

Now, let’s count all distinct line segments:

From the figure:
- AO
- OB
- OC
- OD
- OE
- ON
- DB
- CD

Also, is there AD or CB? No, those aren't drawn.

But wait — CD and DB are connected via D, but not necessarily forming a single segment unless specified.

But we must include only visible segments.

So:

1. AO
2. OB
3. OC
4. OD
5. OE
6. ON
7. DB
8. CD

Are there any others?

- DO and DB make DB already counted.
- CO and OD form CD? Not exactly — CD is from C to D, and it passes through O? No! Wait!

Hold on — C, O, D are not colinear in a straight line. Let's re-express.

Looking closely:

- Point C is on a horizontal line going left-right.
- Point D is just right of O, and B is below it.
- The line from C to D is not passing through O — wait, actually, no.

Wait — C and D are both on the same horizontal line, and O is below them.

But O is the intersection point.

Let’s clarify the structure:

- There is a horizontal line with points: A — O — B
But A is to the left of O, B is to the right of O → So AB is a horizontal line through O.

- Another horizontal line above: C — D → This is parallel to AB? Possibly.

- Then a diagonal line from top-left to bottom-right: passing through C, then O, then E → So line CE passes through C, O, E

- A vertical segment from D down to B → DB

- A ray or segment from O down to N → ON

So now, identify all line segments (with two endpoints):

1. AO
2. OB
3. OC
4. OE
5. ON
6. CD — from C to D (top horizontal)
7. DB — from D to B (vertical)
8. OD — from O to D? But O and D are not directly connected? Wait — is OD a segment?

Wait — D is on the top horizontal line, O is below, so OD is not a segment unless drawn.

But D is connected to B, and O is connected to C, E, N, and possibly D?

Wait — look again: D is a point on the upper horizontal line, and there is a vertical line from D to B, and O is below D.

Is there a segment from O to D? It doesn’t seem so — OD is not drawn.

But O is connected to C (on the diagonal), E (on the diagonal), N, and A and B via AB.

Wait — AB is a line through A, O, B → so segments: AO, OB, AB (but AB includes AO and OB)

But for counting segments, we count individual parts.

So:

List of all line segments (distinct pairs of points connected by a straight line):

1. AO
2. OB
3. OC
4. OE
5. ON
6. CD
7. DB
8. OD — is this present?
- From the diagram: D is connected to B, and O is connected to C, E, N, A, B?
- O to D? No direct line — unless OD is part of a ray.

Wait — O is connected to C via OC, and C to D is a segment, but O to D is not a segment unless drawn.

But C, O, D — are they colinear? No — C and D are on a horizontal line, O is below, so C-O-D is not a straight line.

So OD is not a segment.

Similarly, OC is a segment, OE is a segment.

Now, is CE a segment? No — it's a line passing through C, O, E, so CO and OE are segments.

But CO and OE are already counted.

So far:

- AO
- OB
- OC
- OE
- ON
- CD
- DB

Any more?

What about BO? Already counted as OB

Is there a segment from D to O? No.

Wait — is O connected to D? No.

But D is connected to B, and O is connected to B? Is OB a segment? Yes — O to B is part of AB.

But D to B is a segment.

So total segments:

1. AO
2. OB
3. OC
4. OE
5. ON
6. CD
7. DB

That’s 7.

Wait — what about CO? That’s same as OC.

But is C to O a segment? Yes — OC is drawn.

Yes.

But is CD a segment? Yes — from C to D, horizontal.

And DB — from D to B, vertical.

Is there a segment from O to D? No.

Is there a segment from C to O? Yes — OC

So we have:

- AO
- OB
- OC
- OE
- ON
- CD
- DB

Total = 7

But wait — is AB a segment? It's made of AO and OB, but if A, O, B are colinear, then AB is a single line, but AO, OB, and AB are three segments?

No — when counting line segments, we count all possible distinct segments between two points.

So:

- AO
- OB
- AB (if A to B is considered as one segment)

But in geometry, if three points are colinear: A–O–B, then we have:

- Segment AO
- Segment OB
- Segment AB

But AB is composed of AO and OB.

But in such figures, usually, we count only the smallest segments unless otherwise specified.

But typically, in such problems, each distinct pair of points connected by a line counts as a segment.

So if A, O, B are colinear and connected, then:

- AO
- OB
- AB

But AB is longer and includes AO and OB.

But since AO and OB are drawn separately, and AB is the full line, we usually don’t count AB unless it's explicitly drawn.

But here, the line goes from A to O to B, so AO, OB are segments, and AB is the whole line.

But in terms of line segments, if A, O, B are connected, then:

- AO
- OB
- AB

But AB is not a separate physical segment unless marked.

But in standard practice, we count all segments formed between any two points.

So if A, O, B are colinear and connected, then:

- AO
- OB
- AB

But in this case, A, O, B are on a line, so yes.

But O is the center.

But in most such diagrams, AO, OB are the main segments.

But let's see the diagram: A is on the left, O is center, B is on the right — so AB is a line, and AO and OB are parts.

But AB is not a separate segment unless it's drawn as a single entity.

But since A, O, B are colinear, and AO and OB are drawn, then AB is implied.

But do we count AB as a segment?

In many cases, yes, because it's a valid segment.

But let's think: How many line segments are there?

We need to count all possible line segments between two points that are connected by a straight line.

So list all pairs:

Points: A, B, C, D, E, N, O

Now, which pairs are connected?

- A–O: yes (part of AB)
- O–B: yes
- O–C: yes (diagonal)
- O–E: yes (diagonal)
- O–N: yes
- C–D: yes (top horizontal)
- D–B: yes (vertical)

Is A–B connected? Only through O, but not directly drawn as a single segment? But it's a straight line.

But in such problems, if A–O–B are colinear, then AB is a segment.

Similarly, C–O–E are colinear? Let's see:

- C is on top-left
- O is center
- E is bottom-right

So C–O–E is a straight diagonal line → so C–O, O–E, and C–E are all segments.

Similarly, A–O–B is a straight line → so A–O, O–B, A–B

So we should include:

- AO
- OB
- AB
- CO
- OE
- CE
- CD
- DB
- ON

But CD is separate — from C to D, but C and D are on a horizontal line, but O is not on it.

So C–D is a segment.

Now, is C–D connected through O? No — O is not on CD.

So CD is a separate segment.

Now, is D–O connected? No.

So final list of line segments:

1. AO
2. OB
3. AB
4. CO
5. OE
6. CE
7. CD
8. DB
9. ON

That’s 9 segments.

But AB is redundant if AO and OB are already counted, but technically, AB is a segment.

But in most school-level problems, they ask for distinct line segments based on visible connections.

But often, they count only the segments between adjacent points.

So:

- AO
- OB
- CO
- OE
- ON
- CD
- DB

That’s 7

But C–O and O–E are parts of C–E, so CE is a segment.

But CE is not drawn as a single segment — it's broken at O.

But C–O and O–E are segments.

So we count CO, OE, not CE.

Similarly, A–O, O–B, not AB.

So only the directly connected segments.

So:

- AO
- OB
- CO
- OE
- ON
- CD
- DB

And is OD a segment? No.

Is DN? No.

So 7 segments

But wait — is D connected to O? No.

But D is connected to C and B.

So:

- CD
- DB

Yes.

So total 7 line segments.

But let’s double-check:

1. AO
2. OB
3. CO
4. OE
5. ON
6. CD
7. DB

Yes.

But CO is the same as OC — same segment.

So total = 7

✔ Answer: 7 line segments

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iii) Is there any parallel line, Name them

Look for lines that never meet and are equidistant.

We have:

- Horizontal line AB (through A, O, B)
- Horizontal line CD (through C, D)

These two are horizontal, so likely parallel.

Also, CD is above AB, and both are horizontal.

So AB ∥ CD

Are there any others?

- Diagonal line CE (from C to E) — not parallel to anything else.
- Vertical segment DB — but only a segment, not a line.
- ON — downward ray, not parallel to any.

So only AB and CD are parallel lines.

✔ Answer: Yes, AB and CD are parallel

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iv) Shortest line segment

Look at the segments:

- AO: long
- OB: long
- CO: medium
- OE: medium
- ON: very short
- CD: medium
- DB: medium

ON is the shortest — it’s a small downward segment from O to N.

✔ Answer: ON

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v) ON is perpendicular to

ON is a downward ray from O.

Is it perpendicular to any line?

Look at DB — it’s vertical from D to B.

But ON is not vertical — it’s downward, but not aligned with DB.

Wait — DB is vertical, and ON is also going downward — but is it vertical?

From the diagram, ON appears to be downward, but not necessarily vertical.

But DB is vertical, and ON is pointing downward — but is it perpendicular to something?

Wait — ON is perpendicular to AB?

- AB is horizontal
- ON is downward — if it’s vertical, then yes.

But is ON vertical?

From the diagram, ON is drawn straight down — so it’s vertical.

So ON is vertical, and AB is horizontal → so ON ⊥ AB

Also, CD is horizontal → so ON ⊥ CD

So ON is perpendicular to AB and CD

But the question asks: “ON is perpendicular to” — probably expects one or more.

But likely, AB is the main line.

Since AB and CD are both horizontal, and ON is vertical, it is perpendicular to both.

But perhaps the expected answer is AB

But ON is not necessarily perpendicular to DB — DB is vertical, so ON is parallel to DB, not perpendicular.

So ON is perpendicular to AB and CD

✔ Answer: AB (or CD) — but likely AB

But to be precise: ON is perpendicular to AB and CD

But since the question says "to", maybe just name one.

But let’s see: ON is vertical, AB is horizontal → yes, perpendicular.

So ✔ Answer: AB

(Or CD, but AB is more central)

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vi) How many rays are there in the above figure?

A ray has one endpoint and extends infinitely in one direction.

Look at the figure:

Each arrow indicates a ray.

Let’s count the rays:

1. From A to the left → ray starting at A, going left → ray AA' (leftward)
2. From B to the right → ray starting at B, going right → ray BB'
3. From C to the left → ray starting at C, going left
4. From D to the right → ray starting at D, going right
5. From C to the right → ray from C to D and beyond → but C has two arrows: left and right → so two rays from C
6. From D has two arrows: right and down? Wait — no

Wait — look at the arrows:

- On the top horizontal line:
- Left arrow at C → ray from C to the left
- Right arrow at D → ray from D to the right
- On the bottom horizontal line:
- Left arrow at A → ray from A to the left
- Right arrow at B → ray from B to the right
- On the diagonal line:
- Top-left arrow at C → ray from C to top-left
- Bottom-right arrow at E → ray from E to bottom-right
- On the vertical segment:
- DB is a segment, no arrow at D or B — so not a ray
- ON is a segment, but ends at N — no arrow — so not a ray

Wait — ON has an arrow at N? No — it’s a dot at N, and arrow at the end? Wait — look:

- ON has a dot at N, and the line goes from O to N, with an arrow at N? Or at O?

In the diagram: ON is drawn from O to N, with a dot at N, and the line ends — no arrow.

But the other lines have arrows at ends.

Wait — let’s describe:

- The horizontal line through A and B: has arrows at both ends → so it's a line, but the rays are:
- Ray from A to the left
- Ray from B to the right
- The top horizontal line through C and D: arrows at both ends → so:
- Ray from C to the left
- Ray from D to the right
- The diagonal line from C to E: has arrows at both ends → so:
- Ray from C to top-left
- Ray from E to bottom-right
- The vertical segment DB: no arrows — just a segment
- ON: no arrow — just a segment

But wait — ON is drawn with a dot at N, and the line ends — so it's a segment, not a ray.

So total rays:

1. Ray from A to the left (←)
2. Ray from B to the right (→)
3. Ray from C to the left (←)
4. Ray from D to the right (→)
5. Ray from C to the top-left (↖)
6. Ray from E to the bottom-right (↘)

But C has two rays: one left (on horizontal), one top-left (on diagonal)

Similarly, E has one ray: bottom-right

So total:

- A: 1 ray (left)
- B: 1 ray (right)
- C: 2 rays (left, top-left)
- D: 1 ray (right)
- E: 1 ray (bottom-right)

But D is on the top horizontal — has a ray to the right

C has two rays: one along the horizontal (left), one along the diagonal (top-left)

So total rays:

1. ← from A
2. → from B
3. ← from C (horizontal)
4. ↖ from C (diagonal)
5. → from D (horizontal)
6. ↘ from E (diagonal)

That’s 6 rays

But wait — is the ray from C to the left part of the same line as the ray from D to the right? Yes — they are on the same line, but different rays.

But each arrow represents a ray.

So total 6 rays

But let’s confirm:

- Horizontal line AB: two rays: from A left, from B right
- Horizontal line CD: two rays: from C left, from D right
- Diagonal line CE: two rays: from C top-left, from E bottom-right

So total: 6 rays

✔ Answer: 6 rays

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✔ Final Answers:

i) Name the two intersecting lines?
→ AB and CE

ii) How many line segments in the above figure?
→ 7

iii) Is there any parallel line, Name them
→ Yes, AB and CD

iv) Shortest line segment
→ ON

v) ON is perpendicular to
→ AB (or CD)

vi) How many rays are there in the above figure?
→ 6

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✔ Boxed Answers:

i) AB and CE
ii) 7
iii) AB and CD
iv) ON
v) AB
vi) 6