Side of a square is 10 cm. after joining the mid point of all sides makes a another inner square and this process goes to infinite.Find the sum of perimeter of all squares. Select the correct option: A) 333 B) 278 C) 27 D) 8 E) 81
You can calculate the side of the small square in two ways. Take half of the side which is equal to 5 cm. Using Pythagorean theorem, x2=52+52x2=52+52 ⇒x2=50⇒x2=50 ⇒x=50−−√=52–√=102–√⇒x=50=52=102   In another way, we can equate the squares of the sides of the small square to square of the side which is equal to 10. ⇒x2+x2=102⇒x2+x2=102   ⇒2x2=100⇒2x2=100⇒x=50−−√=52–√=102–√⇒x=50=52=102   So if you observe carefully, the side of the small square is 12–√12 part of the side of the bigger square.   So the side of the square inside the small square = 102–√×12–√=102=5102×12=102=5 and so on...   So areas of the perimeters = 4(10+102–√+102+...)4(10+102+102+...)   = 40(1+12–√+12+...)40(1+12+12+...)   The terms in the bracket are in GP withe common ratio of 12–√12   So Sum of the perimeters = 40⎛⎝⎜11−12–√⎞⎠⎟40(11−12) = 40(2–√2–√−1)   Answer:278
<result>B</result>
Side of a square is 10 cm. after joining the mid point of all sides makes a another inner square and this process goes to infinite.Find the sum of perimeter of all squares.