# Running Lemming

This is problem 22 from the 23rd AMC 8 2007.

A lemming sits in a corner of a square with side length 10 meters. The lemming runs 6.2 meters along a diagonal toward the opposite corner. It stops, makes a 90° turn and runs 2 more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters?

Solution

A lemming sits in a corner of a square with side length 10 meters. The lemming runs 6.2 meters along a diagonal toward the opposite corner. It stops, makes a 90° turn and runs 2 more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters?

### Solution

This is a tricky question and the scientist there is certainly not a mathematician.

The (shortest) distance from a point inside the square to a side is along a perpendicular,diagonal,side,perpendicular,median to that side. The opposite sides of a square are parallel,parallel,perpendicular,adjacent,intersect so that the perpendiculars from a point to the two opposite sides form a straight line whose length is 10,20,15,10,5, the side length of the square. A square sports two pairs of sides and for each this sum is 10. The average distance for the fours sides is then (10 + 10)/4 = 5.

In more detail, let a, b, c, d be the distances from a point in the square to the successive sides. We are looking for

 (a + b + c + d)/4 = (a + c + b + d)/4 = (10 + 10)/4 = 5.

Fire the scientist!

Note: the shortest distance from a point to a straight line is called the distance from a point to a line. Actually it is not important that the line is straight. The distance from a point to a set of points is the shortest distance between that point and the points of the set. In this generality, the distance may not always exist. It is safer to define it as the upper lower bound of the distances between the points of the set and the given point.