ENG  RUS Timus Online Judge
Online Judge
Problems
Authors
Online contests
Site news
Webboard
Problem set
Submit solution
Judge status
Guide
Register
Authors ranklist
Current contest
Scheduled contests
Past contests
Rules
back to board

## Discussion of Problem 1963. Kite

How to solve the problem
Posted by Hidden 27 May 2018 15:16
The problem is very easy. If you solved every geometry problem up to this one you should be able to solve it easily, so you might want to stop reading this.
The axis of symmetry can be drawn in 2 places: through a pair of opposite point or through a pair of midpoints of opposite segments. Why only the midpoints? Because when you "fold" the figure your points must go one ever the other. If the distance is not equal, this is impossible(hint 1).
M A N
__.__
| \  \
|  \  \
|___\__\
B
M is the top-left point
N is the top-right point
This figure for example. A and B are the midpoints. But this is clearly not an axis of symmetry. After the folding, the top-left point will go above the top-right point.
After the folding, the angle relative to the axis of symmetry will be the same. This is a midpoint. The angle with the center in A for example must have 180 degrees;
angle(MAB) + angle(NAB) = 180
angle(MAB) = angle(NAB)
x + x = 180
x = 90
It is now clear that the vector MN must be perpendicular on the vector AB(hint 2)
I think this is more that enough for you to solve the problem/
Good luck!

Edited by author 27.05.2018 15:16

Edited by author 27.05.2018 15:17