Cambridge IGCSE and O Level Additional Mathematics
CHALLENGE Q
16 Starting with an equilateral triangle, a Koch snowflake pattern can be constructed using the following steps:
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Step 1: Divide each line segment into three equal segments. Step 2: Draw an equilateral triangle, pointing outwards, which has the middle segment from step 1 as its base. Step 3: Remove the line segments that were used as the base of the equilateral triangles in step 2.
These three steps are then repeated to produce the next pattern.
Pattern 1
You are given that the triangle in pattern 1 has side length x units.
a Find, in terms of x, expressions for the perimeter of each of patterns 1, 2, 3 and 4 and explain why this progression for the perimeter of the snowflake diverges to infinity.
b Show that the area of each of patterns 1, 2, 3 and 4 can be written as: Pattern
Pattern 2
Pattern 3
Pattern 4
Area
1
3x 2 4
2
x 2 3 3 3x =3 4 4
3
x 2 x 2 3 3 3 9 3x 2 =3 + 12 4 4 4
4
x 2 x 2 x 3 3 3 3 9 27 3x 2 =3 + 12 + 48 4 4 4 4
2
2
Hence show that the progression for the area of the snowflake 8 converges to __ times the area of the original triangle. 5
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