Question 3 - Figure it out (Page 73) - Koch Snowflake - Chapter 4 Class 8 - Exploring Some Geometric Themes (Ganita Prakash II

Transcript

Question 3 - Figure it out (Page 73) Find the perimeter of the shape at the nth step of the sequence. Take the starting equilateral triangle to have a sidelength of 1 unit.Let 𝑷_𝒏 represent the total perimeter. Look closely at what happens to one side of length 1: It gets split into 3 equal pieces of length 1/3. The middle piece is removed, but we add two new pieces of length 1/3 to make the triangle bump. So, a straight line of length 1 is replaced by 4 segments, each measuring 1/3. The new length is 𝟒 × 𝟏/𝟑=𝟒/𝟑. This means at every step, the total perimeter is multiplied by 𝟒/𝟑. Step 0: 3 sides × length 1=3. ( 𝑷_𝟎=𝟑 ) Step 1: 12 sides × length 1/3=4. Or, 3×4/3=4. ( 𝑷_𝟏=𝟒 ) Step 2: 48 sides × length 1/9=48/9=16/3. ( 𝑷_𝟐=𝟏𝟔/𝟑 ) Step 𝒏 Formula: 𝑷_𝒏=𝟑 ×(𝟒/𝟑)^𝒏