BdMO Online Forum

Problem 9:
A square has sides of length $2$. Let $S$ is the set of all line segments that have length $2$ and whose endpoints are on adjacent side of the square. Say $L$ is the set of the midpoints of all segments in $S$. Find out the area enclosed by $L$.

\[\sqrt{2}\] is the length of the segments by the mid-points.
So the area is 2...

photon wrote:\[\sqrt{2}\] is the length of the segments by the mid-points.
So the area is 2...

How have you got that $L$ will be a square of side length $\sqrt 2$ ?

I just used pythagorean theorem.
Here the value of 2 sides of right angle is 1(for each right angled triangle)
So hipotenuse is \[\sqrt{2}\]

How can you be sure that $L$ is a square? See the image below. The Blue lines are some segments that have length $2$ and whose endpoints are on adjacent side of the square. $S$ is the set of all such segments. Join the midpoints of the segments and find out the shape of the area enclosed by the midpoints

oops
Thanks,brother. give some hint?

draw segments and guess the shape. try to prove your assumption... try it yourself, it will be fun

is it possible to solve it without calculas?

i think it'll be a circle whose radius is $\frac {1}{\sqrt 2}$
so,the area will be $\frac {\pi}{2}$

Hint Solution