Mind Your Decisions, I'm Presh Talwalkar.
At a rectangular table are four mathematicians A, B, C and D.
They sit at the following distances to the corners of the table.
A salt shaker S is placed on the table.
Consider the distance from each person to the saltshaker and take the sum of these distances.
At some point, the sum of these distances will be a minimum value.
The question is: What is the minimum value of SA + SB + SC + SD?
This problem has tremendous practical applications.
You could imagine the four points are either towns or railway stations and you want to calculate
the least amount of roads or tracks that you would need to connect these towns or railway stations.
You could also imagine the four points are friends and they want to calculate the most environmentally friendly point to meet,
that is the one that minimizes the sum of the distances they have to travel.
You can really use this problem for many applications in teamwork.
I adapted this problem from a Sri Lanka Olympiad and I thank Chanul for the suggestion.
Pause the video if you'd like to give this problem a try, and when you're ready, keep watching to learn how to solve this problem.
Imagine we have four points in the plane that form a convex quadrilateral.
We want to determine the point S such that the sum of distances to these four points is a minimum value.
We will prove the rather fascinating result: S will be at the intersection of AC and BD.
First let's prove why this is true.
Consider the segment AC
For any point X in the plane, we can connect AX and XC. Let's consider those distances.
Well, by the triangle inequality AC will be less than or equal to AX + CX.
Then, AC is equal to AS + CS.
We will put this inequality to the side and for the same point X, we will consider the line segment BD.
We will do a very similar sort of argument.
By the triangle inequality, BD is less than or equal to BX + DX.
BD is equal to BS + DS. We thus have another inequality.
Let's put these two inequalities together by adding them. What happens?
Well on the left hand side, we get the sum of the four distances from each of these points to S,
and on the right hand side, we have the sum of distances from some other point X.
And we have shown that S is a minimum value; no other point can be less than that.
Therefore, this is a minimal value.
Let's use this result to solve the problem.
We will first determine where S is by connecting AC and BD.
Their intersection point will be S.
From here, we need to calculate AS + SC + BS + SD.
We'll do that by using right triangles.
First, let's calculate the length of AC.
We have one right triangle here. One leg will be equal to 22 and the other leg will be equal to 60.
Therefore we can calculate the length of the hypotenuse using the Gougu theorem
and we get 2 times the square root of 1021.
This will be equal to AS + CS.
We can then do the same sort of calculation to determine the length of BD.
We have a right triangle where one leg is 25 and the other leg is 40.
We can therefore calculate the length of BD using the Gougu theorem
and we get 5 times the square root of 89.
This will be equal to BS + TS.
We then can calculate the sum of the four distances and that will be equal to
2 times the square root of 1021, plus 5 times the square root of 89
and that's approximately 111.08 and that's our answer.
It's a rather fascinating optimization problem with an elegant geometric solution.
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As always, thanks for watching and thanks for your support.