"I don't cut corners", he announced proudly. "I do everything by the book!". "Good for you", I replied with a grin. I then continued, "actually, did you know that by cutting corners you can save up to 29%?"
He looked at me with an agitated and surprised look. "Where did you come up with that? T W E N T Y N I N E P E R C E N T!". He sounded angry. I guess he wasn't expecting me to quantify cutting corners. For him, the phrase `cutting corners' is associated with being hasty. In such cases, it often isn't clear how much you save. Indeed they say that `haste makes waste', so even if you save on work or materials, compromising quality can be damaging in the long run.
But I was not thinking of our joint work when I brought up my 29%. I rather had a pleasant picture such as this in mind:
We see such corners in our school grounds, cities, parks and countryside. It is so often the case, that walking paths or roads intersect at right angles. When I walk, I always seek to `cut across'. Perhaps in a previous life I was a bird.
As a concrete illustration, see this Google Maps satellite image of a playing field in Urbana-Champaign, IL, USA:
Say you are at the southwest corner and wish to reach the northeast corner. According to Google, this distance of 0.5 miles, about 800 meters would take about 10 minutes to walk. But wouldn't it be better to cut across?
Walk in a north-east direction as opposed to first east and then north. This would save you about 29%, so you would walk for about 7 minutes instead of 10.
You may now ask where 29% comes from?
When you have a 90 degree corner, you have a triangle. A right triangle. Then when it comes to right triangles, you have one of the most celebrated elementary classic mathematical results of all times: The Pythagoras Theorem. See this short video:
Q: Is the yellow triangle in the video a right triangle?
A: Yes, it has a 90 degree angle. This means that the side opposing the right angle is called the hypotenuse. The hypotenuse is always the longest side of a right triangle.
Q: The video shows that the quantity of liquid that fits in the square created by the hypotenuse is exactly the same as the quantity of liquid fitting in the sum of the other squares. How would you describe this in words?
A: This is the Pythagoras Theorem:
The square of the hypotenuse equals the sum of the squares of the two sides.
Q: If we denote the lengths of the sides by A and B, and the length of the hypotenuse by C, how would we describe the relationship between A, B and C in an equation?
A: Here are a few alternatives:
Or if you apply square root to both sides,
Or if you try to find A in terms of C and B,
So you see that you can represent C in terms of the other sides, or A in terms of the others sides. Try to see if you can find an expression for B in terms of C and A, it works just the same as the above. You can also try to see how the theorem works for typical Pythagorean triples, such as the famous triple, A = 3, B = 4, C = 5 or A = 5, B = 12, C = 13.
Maybe even draw triangles of such dimensions (or proportional dimensions) with your kids. If you want to review it a bit, here is a useful elementary summary of Pythagoras by "Math is Fun".
So having reviewed Pythagoras, it is now a rather simple matter to see our 29%. For this we should first agree that a right triangle that is also isosceles maximizes the percent savings. Having such a triangle means we cut the corner at an angle of 45 degrees.
One way to see this is to design a function that measures our relative saving. For example,
This function considers a right triangle with A = 1 and B = x. What is then the length of the hypotenuse? Where do you see it in the function? The denominator is the distance we walk without cutting the corner. That is 1 + x. The numerator is then the distance we save by cutting.
We could have designed the function with other values for A as opposed to 1. But it doesn't really matter, the point is that when x=1, it holds that A=B. Here is a plot of the function for x in the range of 0 to 10.
When x is either very small or very large, there is hardly a corner to cut and hence there isn't much saving, but when x = 1, we have a 45 degree angles in our triangle (it is an isosceles right triangle) and as you can see we save about 29%. The function f(x) indeed attains a maximum at x=1. This value is at,
Indeed when dealing with such neat triangles, the square root of 2 always pops up. We discussed it when we surveyed other irrational numbers, and we'll surely explore it more in the future.
As I finished telling him about the origins of my 29% he looked at me with appreciation. "Thank you for stopping our silly debate and reminding me how much I like geometry and mathematics", he said.
I smiled and thought to myself, "what a nice thing of him to say". I was then reminded about how much I appreciate him. "Why?" I thought to myself. I pondered about it for a while and then came to a clear conclusion. I really appreciate him because he doesn't cut corners.