3D-Shapes Symmetry Patterns-Term2 Worksheet | PDF | Polytopes ... - Free Printable

Let's solve each question in the worksheet step by step and explain the reasoning.

---

Level 1

#### Q1. How many lines of symmetry does this equilateral triangle have?

An equilateral triangle has all sides equal and all angles equal (60°).
A line of symmetry is a line that divides the shape into two identical halves.

- In an equilateral triangle, you can draw a line from each vertex to the midpoint of the opposite side.
- Each such line is a line of symmetry.

✔ Answer: 3 lines of symmetry

---

#### Q2. Observe the pattern and continue it.

Let’s analyze each sequence:

---

1. 9, 12, 15, 18, ____, ____, ____

- Pattern: Increasing by 3
- 9 → 12 (+3), 12 → 15 (+3), 15 → 18 (+3)
- So next: 21, 24, 27

✔ Answer: 21, 24, 27

---

2. 20, 40, 60, 80, ____, ____, ____

- Pattern: Increasing by 20
- 20 → 40 (+20), 40 → 60 (+20), etc.
- Next: 100, 120, 140

✔ Answer: 100, 120, 140

---

3. 1, 2, 4, 7, 11, ____, ____, ____

- Look at differences:
- 2 – 1 = 1
- 4 – 2 = 2
- 7 – 4 = 3
- 11 – 7 = 4
- Differences increase by 1: +1, +2, +3, +4 → next difference is +5
- 11 + 5 = 16
- Then +6 → 16 + 6 = 22
- Then +7 → 22 + 7 = 29

✔ Answer: 16, 22, 29

---

4. 99, 88, 77, 66, ____, ____, ____

- Decreasing by 11 each time
- 99 → 88 (-11), 88 → 77 (-11), 77 → 66 (-11)
- Next: 55, 44, 33

✔ Answer: 55, 44, 33

---

5. 1, 2, 4, 8, ____, ____, ____

- Powers of 2:
- 2⁰ = 1
- 2¹ = 2
- 2² = 4
- 2³ = 8
- 2⁴ = 16
- 2⁵ = 32
- 2⁶ = 64

✔ Answer: 16, 32, 64

---

6. ABC, XYZ, DEF, UVW, ____, ____, ____

- Alternating sequences:
- First group: ABC → DEF → GHI? But let’s see:
- ABC → DEF (skip 2 letters after C: D,E,F)
- XYZ → UVW (backwards: X→U, Y→V, Z→W — decreasing by 3)
- So pattern might be alternating forward and backward?

Wait, let's list them:
1. ABC (A,B,C)
2. XYZ (X,Y,Z) ← last three letters
3. DEF (D,E,F) ← next after C
4. UVW (U,V,W) ← before X

So the pattern seems to be:
- ABC (start)
- XYZ (end)
- DEF (next block)
- UVW (previous block)

But this is not consistent.

Alternatively, think of it as:
- ABC → DEF → GHI → JKL (forward alphabet)
- XYZ → UVW → TSR → QPO (backward)

But here we have:
- ABC
- XYZ
- DEF
- UVW

So it's like:
- Group 1: ABC (A–C)
- Group 2: XYZ (X–Z)
- Group 3: DEF (D–F)
- Group 4: UVW (U–W)

Now look at starting letters:
- A, X, D, U

Not clear.

Wait — maybe it's two interleaved sequences:

Sequence 1: ABC, DEF, GHI, JKL... (every 3 letters forward)
Sequence 2: XYZ, UVW, TSR, QPO... (reverse order)

But here:
- ABC (first)
- XYZ (last)
- DEF (second)
- UVW (before last)

So perhaps:
- First: ABC
- Second: XYZ
- Third: DEF
- Fourth: UVW
- Fifth: GHI
- Sixth: TSR
- Seventh: JKL
- Eighth: QPO

So the pattern alternates between:
- Forward blocks: ABC, DEF, GHI, JKL...
- Backward blocks: XYZ, UVW, TSR, QPO...

So:
- After UVW comes TSR (T,U,V → T,S,R backwards)
- Then GHI (after DEF)
- Then QPO (after TSR)

Wait, but the order is:
1. ABC
2. XYZ
3. DEF
4. UVW
5. ??? (should be next forward: GHI)
6. ??? (next backward: TSR)
7. ??? (next forward: JKL)
8. ??? (next backward: QPO)

But since the pattern is alternating:
- Odd positions: ABC, DEF, GHI, JKL...
- Even positions: XYZ, UVW, TSR, QPO...

So:
- 5th: GHI
- 6th: TSR
- 7th: JKL
- 8th: QPO

✔ Answer: GHI, TSR, JKL

But wait — the sequence given is:
1. ABC
2. XYZ
3. DEF
4. UVW
5. ___
6. ___
7. ___

So:
- Position 1: ABC
- Position 2: XYZ
- Position 3: DEF
- Position 4: UVW
- Position 5: GHI (next forward)
- Position 6: TSR (next backward)
- Position 7: JKL (next forward)

So the pattern is:
- Forward: ABC, DEF, GHI, JKL...
- Backward: XYZ, UVW, TSR, QPO...

Thus:
✔ Answer: GHI, TSR, JKL

---

7. 10AB, 20BC, 30CD, 40DE, ____, ____, ____

Look at parts:
- Numbers: 10, 20, 30, 40 → increasing by 10 → next: 50, 60, 70
- Letters: AB, BC, CD, DE → next: EF, FG, GH

So:
- 50EF
- 60FG
- 70GH

✔ Answer: 50EF, 60FG, 70GH

---

#### Q3. Write the number of faces, edges, and vertices of the following shapes:

a. Cube

- Faces: 6 (top, bottom, front, back, left, right)
- Edges: 12 (each face has 4 edges, but shared)
- Vertices: 8 (corners)

✔ Faces = 6, Vertices = 8, Edges = 12

b. Cylinder

- Faces: 3 (2 circular bases + 1 curved surface)
- Edges: 2 (the circular edges — top and bottom)
- Vertices: 0 (no sharp corners)

Note: In 3D geometry, a cylinder has:
- 2 flat faces (bases)
- 1 curved face (lateral surface)
- No vertices (since no corners)
- 2 edges (where the base meets the lateral surface)

✔ Faces = 3, Vertices = 0, Edges = 2

---

Level 2

#### Q1. How many lines of symmetry does the following figures have?

Figure 1: Circle

- A circle has infinite lines of symmetry.
- Any line passing through the center divides it into two equal halves.

✔ Answer: Infinite lines of symmetry

Figure 2: Arrow-shaped figure (like a cross with arrows pointing out)

This looks like a symmetric cross with four arrowheads pointing in cardinal directions.

It has rotational symmetry and reflection symmetry.

Let’s analyze:
- It has vertical line of symmetry (left-right mirror)
- Horizontal line of symmetry (up-down mirror)
- Diagonal lines? Probably not, because the arrows are straight.

But if the shape is symmetric across both vertical and horizontal axes, then:
- Vertical line: yes
- Horizontal line: yes
- Diagonals? Only if the arrows are symmetric diagonally — but typically in such diagrams, they are not.

Assuming it's symmetric only along vertical and horizontal axes.

So:
- 1 vertical
- 1 horizontal

✔ Answer: 2 lines of symmetry

But wait — sometimes such shapes have more.

If the shape is like a plus sign with arrowheads, and the arrowheads are identical in all directions, then it may have 4 lines of symmetry:
- Vertical
- Horizontal
- Two diagonals

But looking at the diagram (as described): it's like a "cross" with arrows pointing left/right/up/down, and each arm has a rectangular extension.

But unless the arms are symmetrical diagonally, it likely has only vertical and horizontal symmetry.

However, if the shape is rotationally symmetric by 90°, it may also have diagonal symmetry.

But most commonly, such a shape (like a compass rose) has 4 lines of symmetry: vertical, horizontal, and two diagonals.

But in this case, the shape is drawn with rectangular arms, so it might have only 2 lines of symmetry.

But let’s assume it’s designed symmetrically.

Looking at standard interpretations:
- If it's a square-like cross with arrows, and all arms are identical, then it has:
- 4 lines of symmetry: vertical, horizontal, and two diagonals.

But in the image description, it's shown as a central diamond-like shape with arrows on ends, resembling a four-pointed star or cross.

But without seeing the exact image, based on typical school-level problems, this kind of figure usually has:

✔ Answer: 4 lines of symmetry

But let’s reconsider: if the figure is made of two overlapping rectangles (horizontal and vertical), forming a cross, then:
- Symmetry: vertical, horizontal → 2 lines
- Diagonals? Only if the arms are square, but if they're rectangular, diagonals won't work.

So probably 2 lines of symmetry.

But the common version of this shape (like a "diamond" with arrows) often has 4 lines.

Given ambiguity, but based on standard practice:

✔ Most likely answer: 4 lines of symmetry

But to be safe, if it's a cross with rectangular arms, it has 2 lines of symmetry.

But since the arrows are outward and the shape appears balanced, and often such figures are designed for 4-fold symmetry, I'll go with:

✔ Answer: 4 lines of symmetry

---

✔ Final Answers Summary:

---

#### Level 1

Q1.
Lines of symmetry in equilateral triangle: 3

---

Q2.
1. 21, 24, 27
2. 100, 120, 140
3. 16, 22, 29
4. 55, 44, 33
5. 16, 32, 64
6. GHI, TSR, JKL
7. 50EF, 60FG, 70GH

---

Q3.
a. Cube:
- Faces = 6
- Vertices = 8
- Edges = 12

b. Cylinder:
- Faces = 3
- Vertices = 0
- Edges = 2

---

#### Level 2

Q1.
- Circle: Infinite lines of symmetry
- Arrow-cross shape: 4 lines of symmetry

---

✔ All answers explained and solved. Let me know if you'd like this formatted as a filled-in worksheet!