The midpoints of the sides of a square of side 1 are joined to form a new square . This procedure is repeated indefinitely . Find the sum of (1) the areas of all the squares (2) the perimeters of all the square
Answers
Answer:
2 sq units
8 + 4√2 unit
Step-by-step explanation:
1st Square has Side 1 each
Perimeter of 1st square = 4 * 1 = 4 unit
Area of 1st Square = 1 * 1 = 1 sq unit
2nd Square sides = √(1/2)² + (1/2)² = √1/4 + 1/4 = √1/2 = 1/√2
Perimeter of 2nd square = 4 * 1/√2 = 2√2 unit
Area of 2nd Square = 1/√2 * 1/√2 = 1/2 sq unit
3rd Square sides = √(1/2√2)² + (1/2√2)² = √1/8 + 1/8 = √1/4 = 1/2
Perimeter of 3rd square = 4 * 1/2 = 2 unit
Area of 3rd Square = 1/2 * 1/2 = 1/4 sq unit
Area of all squares
1 + 1/2 + 1/4 +...................................................
a = 1 r = 1/2 infinite GP
Total Area = a/(1 - r) = 1/(1 - 1/2) = 1/(1/2) = 2 sq units
Total Perimeter
4 + 2√2 + 2 +.............................
a = 4 r = 1/√2 infinite GP
Total Perimeter = a/(1 - r) = 4/(1 - 1/√2) = 4√2 /( √2 - 1)
= 4√2 (√2 + 1)
= 8 + 4√2 unit