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.

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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

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

Let’s analyze each sequence:

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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

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

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

✔ Answer: 100, 120, 140

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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

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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

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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

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

- Alternating sequences:
- First group: ABC → DEF → GHI (next)
- Second group: XYZ → UVW → TSR? Wait — let's look carefully.

But observe:

- ABC (first three letters)
- XYZ (last three letters)
- DEF (next after ABC)
- UVW (before XYZ)

So it seems like alternating between:
- Forward: ABC → DEF → GHI
- Backward: XYZ → UVW → TUV? But UVW is before XYZ.

Wait: XYZ → UVW → TSR? Let's check:

- XYZ → UVW: X→U (back by 3), Y→V (back by 3), Z→W (back by 3)
- So reverse order with step back by 3.

But actually:

- ABC → DEF: A→D (+3), B→E (+3), C→F (+3)
- XYZ → UVW: X→U (-3), Y→V (-3), Z→W (-3)

So pattern: forward +3, then backward -3, repeat?

So next should be: GHI (DEF +3) → then TSR (UVW -3)

But wait: after UVW, what comes?

Sequence: ABC, XYZ, DEF, UVW, ?, ?, ?

So positions:
1. ABC (start)
2. XYZ (end)
3. DEF (next start)
4. UVW (previous to XYZ)
5. ? → likely GHI (after DEF)
6. ? → likely TSR (before UVW)
7. ? → likely IJK (after GHI)

But better to see as two interleaved sequences:

- Sequence 1 (forward): ABC, DEF, GHI, JKL...
- Sequence 2 (backward): XYZ, UVW, TSR, QPO...

So the pattern alternates:

1. ABC
2. XYZ
3. DEF
4. UVW
5. GHI
6. TSR
7. JKL
8. QPO

✔ Answer: GHI, TSR, JKL

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

- Numbers: 10, 20, 30, 40 → increasing by 10 → next: 50, 60, 70
- Letters: AB, BC, CD, DE → each pair moves forward by one letter
- AB → BC (B→C), BC → CD (C→D), CD → DE (D→E), so next: EF, FG, GH

So:
- 50EF, 60FG, 70GH

✔ Answer: 50EF, 60FG, 70GH

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#### 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; total 12 unique edges)
- Vertices: 8 (corners)

✔ Answer:
- Faces = 6
- Vertices = 8
- Edges = 12

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##### b. Cylinder

- Faces: 3
- 2 circular bases (top and bottom)
- 1 curved surface (lateral face)
- Edges: 2 (the circular boundaries — though not straight, in elementary math, often counted as 2 edges)
- Vertices: 0 (no sharp corners)

Note: In some definitions, cylinders have no vertices, but for school level, sometimes they consider the circles as edges and no vertices.

✔ Answer:
- Faces = 3
- Vertices = 0
- Edges = 2

> Note: Some curricula may define cylinder differently, but standard elementary definition:
> - Faces: 3 (2 flat, 1 curved)
> - Edges: 2 (top and bottom circular edges)
> - Vertices: 0

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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

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

The figure looks like a symmetric arrowhead cross, with four arrowheads pointing in four directions (up, down, left, right), forming a symmetrical star-like shape.

This shape has rotational symmetry and reflection symmetry.

Let’s analyze:

- It has horizontal line of symmetry (top-bottom)
- Vertical line of symmetry (left-right)
- Diagonal lines? Probably not — unless the arms are symmetric diagonally.

But if it's a square-based shape with arrows on each side (like a compass rose), then it likely has:

- Horizontal symmetry
- Vertical symmetry
- Two diagonal symmetries

But looking at the drawing (though not visible here), based on typical such problems, if it's a four-pointed arrow cross, it usually has 4 lines of symmetry:

- One vertical
- One horizontal
- Two diagonals

But if it's only symmetric along vertical and horizontal, then 2.

However, since it’s a cross with arrowheads on all four sides, and assuming it's perfectly symmetrical, then:

✔ Answer: 4 lines of symmetry

> If the shape is like a square with arrows on each side, pointing outward, and symmetric, then yes, it has 4 lines of symmetry.

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✔ Final Answers Summary:

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#### Level 1

Q1: 3 lines of symmetry

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

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#### Level 2

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

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