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Geometry - Symmetry and Patterns

Grade 4ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Symmetry exists when one shape becomes exactly like another if you flip, slide or turn it. The imaginary line where you could fold the image and have both halves match exactly is called the Line of Symmetry.

An isosceles triangle with a vertical line of symmetry passing through the top vertex and the midpoint of the base.
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A Rectangle has 22 lines of symmetry: one horizontal and one vertical. Unlike a square, the diagonals of a rectangle are not lines of symmetry because folding along them does not align the corners.

A rectangle showing two lines of symmetry: one horizontal and one vertical.
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Reflection symmetry creates a 'mirror image'. In a reflection, every point of the original object is at the same distance from the mirror line as the corresponding point of the reflected image.

Two shapes facing each other across a central vertical line, illustrating reflection symmetry.
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Patterns are sequences that repeat based on a specific rule. Geometric patterns often involve shapes rotating, changing size, or alternating in a predictable way.

A pattern sequence consisting of circle, square, circle, square.

📐Formulae

Number of lines of symmetry in a regular polygon=n\text{Number of lines of symmetry in a regular polygon} = n (where nn is the number of sides)

Lines of symmetry in a Square=4\text{Lines of symmetry in a Square} = 4

Lines of symmetry in an Equilateral Triangle=3\text{Lines of symmetry in an Equilateral Triangle} = 3

Lines of symmetry in a Circle=∞\text{Lines of symmetry in a Circle} = \infty (Infinite lines passing through the center)

Pattern Rule (Addition):Termn=Termn−1+d\text{Pattern Rule (Addition)}: \text{Term}_{n} = \text{Term}_{n-1} + d

💡Examples

Problem 1:

Determine the number of lines of symmetry for a regular hexagon and describe where they are located.

Solution:

Step 1: Identify the number of sides in a regular hexagon. A hexagon has n=6n = 6 sides. Step 2: Use the rule for regular polygons, which states that the number of lines of symmetry equals the number of sides. Step 3: Count the lines. There are 33 lines passing through the opposite vertices (corners) and 33 lines passing through the midpoints of opposite sides. Total lines of symmetry = 66.

Explanation:

Since a regular hexagon has equal sides and angles, any line passing through the center to opposite vertices or midpoints will divide it into two identical halves.

Problem 2:

Identify the rule and find the next two terms in the pattern: 3,6,12,24,…3, 6, 12, 24, \dots

Solution:

Step 1: Look at the relationship between the first and second terms: 3×2=63 \times 2 = 6. Step 2: Check if this rule applies to the rest: 6×2=126 \times 2 = 12 and 12×2=2412 \times 2 = 24. The rule is 'multiply by 22'. Step 3: Calculate the next term: 24×2=4824 \times 2 = 48. Step 4: Calculate the term after that: 48×2=9648 \times 2 = 96. The next two terms are 4848 and 9696.

Explanation:

This is a growing pattern where each subsequent number is double the previous number.

Problem 3:

Identify the number of lines of symmetry in an isosceles triangle where only two sides are equal.

Isosceles triangle with one vertical line of symmetry.

Solution:

An isosceles triangle has 11 line of symmetry.

Explanation:

In an isosceles triangle, only the line passing through the vertex between the equal sides and the midpoint of the opposite base divides it into two identical halves. Lines through the other vertices do not result in matching halves because the side lengths are unequal.

Problem 4:

Complete the following number pattern and identify the rule: 2,5,8,11,…2, 5, 8, 11, \dots

A flow diagram showing the addition of 3 to each consecutive number in the pattern 2, 5, 8, 11, 14, 17.

Solution:

The next two terms are 1414 and 1717. The rule is 'Add 33'.

Explanation:

Check the difference between consecutive terms: 5−2=35 - 2 = 3 8−5=38 - 5 = 3 11−8=311 - 8 = 3 Since the difference is constant, we add 33 to the last term to find the next: 11+3=1411 + 3 = 14 14+3=1714 + 3 = 17