Geometry-9 (CAT-2001)

A square, whose side is 2 meters, has its corners cut away so as to form an octagon with all sides equal. Then the length of each side of octagon, in meters is:
(1)√2/√2+1      (2) 2/√2+1        (3) 2/√2-1        (4) √2/√2-1
Solution follows here:

Solution:

ABCD is square and PQRSTUVW is octagon with equal sides constructed with in the square.

In order to have octagon with equal sides, we should have AP=QB=BR=SC=CT=….

Let AP=AW=x;

Let PQ = y; as all sides of octagon are equal, QR=RS=ST=TU=UV=VW=WP=y;

As the length of side of square is 2,

2x+y = 2         ------(1)

Applying Pythagoras to the right triangle APW, y² = x²+x² = 2x² 

=>  y= x√2      ------(2)

Solving (1) and (2):

2x+x√2 = 2 => x = 2/(2+√2) = √2/(√2+1)

=>  y = x√2 = 2/(√2+1)

Answer (2)