Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Koch Snowflake. Class 8

Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Koch Snowflake. Class 8

Question 1.

Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Koch Snowflake. Class 8

Solution:

Note : Do as directed as drawn in the textbook.

Question 2.

Find the number of sides in the nth step of the shape sequence that leads to the Koch Snowflake. Class 8

Solution:

In step 1, number of sides = 12 = 3 × 4.

In step 2, number of sides = 48 = 3 × 16

= 3 × 4².

In step 3, number of sides = 192 = 3 × 64

= 3 × 4³, ...

In step n, number of sides = 3 × 4ⁿ.

Question 3.

Find the perimeter of the shape at the nth step of the sequence. Take the starting' equilateral triangle to have a sidelengfh of 1 unit. Class 8

Solution:

Length of each side at step 1 = [latex]\frac{1}{3}[/latex] and

number of sides at step 1 = 12 = 3 × 4.

So, perimeter at step 1 = 3 × ([latex]\frac{4}{3}[/latex]) units.

Length of each side at step 2 = [latex]\left(\frac{1}{3}\right)^2[/latex] and number of sides at step 2 = 48 = 3 × 4².

So, perimeter at the nth step = 3 × 4² × [latex]\left(\frac{1}{3}\right)^2[/latex] = 3 × [latex]\left(\frac{4}{3}\right)^2[/latex] units.

Length of each side at step 3 = [latex]\left(\frac{1}{3}\right)^3[/latex] and number of sides at step 3 = 192 = 3 × 4³.

So, perimeter at step 3 = 3 × 4³ × [latex]\left(\frac{1}{3}\right)^3[/latex]

= 3 × [latex]\left(\frac{4}{3}\right)^3[/latex] units.

Thus, perimeter of the step at the nth step = 3 × [latex]\left(\frac{4}{3}\right)^n[/latex] units.