I’ve always heard the shortest distance between 2 points is a straight line. But if thats true then how does the A^2 and B^2 parts of a triangle factor into that? Wouldn’t they also be the shortest distance between 2 points?
Edit: I’m trying to put this into walking perspective, actually hay stacking. Say I have a bale of hay in the very center of the field. I need to take it to the very southwest corner of the field. You’d think just taking the hay in a straight line (directly southwest from the center) would be faster than going to the very south of the field, and then the very west of the field until you reach the stack. Wouldn’t both ways you go be the same distance? So therefore the shortest distance to between 2 points isn’t JUST a straight line, it’s also a right angle?
In: Mathematics
It may be easier to imagine this with A and B being 1:
A = 1
B = 1
C = ?
If you move right one and up 1, you have moved 2. If you instead move diagonal you moved more than 1, but less than 2. Why? Because to calculate C, you have to find the number that multiplies by itself to equal (A * A) + (B * B), in this case you have to find a number that multiplies by itself to equal 2.
That number can’t be 1, because 1 * 1 = 1, and it can’t be 2, because 2 * 2 = 4 (which is larger than the target of 2), so it *must* be larger than 1, but less than 2.
Therefore the only number that fits this equation, is roughly 1.4142, because 1.4142 * 1.4142 = 1.99996, rounding to 2.
And hopefully you would agree that 1.4142 is *less than* the target number of 2, making the shortest distance A to C instead of A to B then B to C.
As to why the above works that way, that’s simply how right angles work. You would have to change some laws of physics to change it. The Pythagorean theorem is simply a mathematical function that allows us to calculate this.
I think you may have an initial confusion.
The shortest distance between two points is a straight line. In your example of going from the middle of the field to a corner, it’s the straight line.
Going down to the bottom, then turning 90 degrees would be longer.
The Pythagorean theorem *does not* say that the distance south then over would be the same as the distance diagonal. I think you maybe think this is what it says? But it doesn’t. That would be a+b=c, which is not true.
Instead it says a^2 + b^2 = c^2 . That’s not the same thing, and doesn’t mean the lengths of a and b are the same as the length of c
There have been some very acute explanations here, but for those of us who are too obtuse to get it there has to be an easier way, right?
To use your hay bale example and make sense, pretend the bale is on the northeast corner and the wagon is on the southwest corner. The field is a perfect square 600 feet to a side. If the field hasn’t been cut, you have to walk 600 feet from the northeast corner to the southeast corner (A to B), then another 600 feet to the southwest corner (B to C) for a total of 1200 feet.
If the field has been cut, you can walk a straight line from the northeast to the southwest (A to C). The Pythagorean theorem is how you find out how long *that* walk is based on how long the sides of the field are.
It is precisely *because* a straight line is the shortest distance between two points that the Pythagorean Theorem can be used to calculate a side of a right triangle.
Draw a right angle. (that part is important).
You can make one leg tiny and the other long, it doesn’t matter.
Connect the ends of those two lines with a straight line.
That straight line is the shortest distance between those two points!
And the Pythagorean Theorem gives us the means to calculate that distance.
And actually, with ***any*** triangle you can draw, any combination of two of its lines will be longer than the third.
The a^2 and b^2 aren’t part of the triangle. The Pythagorean Theorem simply states that, for any right triangle, if you drew three squares, each with sides the length of the three sides of the triangle, that the area of the biggest square would be equal to the areas of the two smaller squares added together.
There are three point in a triangle. A squared is AB, B squared is BC, and C squared is AC.
The Pythagorean theorem is meant to calculate the shortest distance between two points (the hypotenuse) by way of the two sides that basically on their own are the X distance of the points and Y distance of the points. When you square those values and add them, you get the square of the shortest line between the two points!
So, yes, A and B are the distance between 2 points each. But, A and B represent 2 separate lines. A is between point A1 and A2. B is between A2 and B1. Pythagorean theorem is trying to find the line that exists between A1 and B1, a previously unmeasured segment.
Edit: Daw two lines that form a right angle, label the points A1, A2, B1. Then draw the third line between A1 and B1 — this is the line you’re trying to find the length of.
Hope this makes sense. Basically your question is asking about something the theorem isn’t about.
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