Why Isn't the Hypotenuse the Sum of the Other Two Sides?

[Shawn Garsed]

Lately this has been bothering me, I hope you can understand the point I'm trying to make. On a side note, this topic maybe more philosophical than mathematical. It comes down to this. In a right triangle, why is the length of the hypotenuse not equal to the length of the adjacent side plus the length of the opposite side. When I think of the adjacent and opposite side as being the hypotenuse broken down into a vertical and a horizontal displacement where the hypotenuse is the total displacement, it seems logical that the hypotenuse is equal to the adjacent side plus the opposite side, since they both cover the same ground. Now I know about the Pythagorean Theorem, it's just that in my mind the above story makes perfect sense.

P.S.
This is an entirely different question, but I didn't want to start a new topic for it. Do you guys know some websites where they have algebra exercises, I like to test my algebra 'skills' once in a while too keep them fresh.

 

[zgozvrm]

You're confusing geometry and physics.

Let's use your example of a right triangle. Let's call the vertical leg of the triangle side A, the horizontal side of the triangle side B, and the hypotenuse side C.

We know, by the Pythagorean Theorem, that C² = A² + B².

But, when dealing with vectors (let's assume side A represents a vector pointing downward and side B represents a vector pointing to the right), we say that C = A + B. In this case (since we're talking about vectors), we're looking at C as a the resultant vector of 2 displacement vectors A & B. This vector will have a length that is the square root of the sum of the squares of the magnitudes of vectors A & B. So, we still have C² = A² + B². But, being a vector, we also know the direction of vector C (which would be at a diagonal direction down, and to the right).

So, vector addition involves finding both the magnitude and direction, whereas in geometry, we're only dealing with magnitudes.

 

[HallsofIvy]

Even if you think of this as "physics" you are adding vectors incorrectly.

 

[zgozvrm]

Even if you think of this as "physics" you are adding vectors incorrectly.

Note that both the OP and I are referring to only 2 vectors at right angles, so how is it they are being added incorrectly, other than leaving out the calculation for the direction, which I referred to (but didn't calculate, or give the formula for) in my last post?

 

[gb7nash]

Lately this has been bothering me, I hope you can understand the point I'm trying to make. On a side note, this topic maybe more philosophical than mathematical. It comes down to this. In a right triangle, why is the length of the hypotenuse not equal to the length of the adjacent side plus the length of the opposite side.

Perhaps an easier way of thinking about it is this. Label the vertices that contain the hypotenuse A and B, and let C be the vertex with the 90 degree angle. As we all should know, the shortest distance between A and B is a straight line (which happens to be the hypotenuse). Now follow A to C and C to B. This is NOT a straight line and hence the length of the hypotenuse is less than the length of the two other sides summed together.

 

[zgozvrm]

Perhaps an easier way of thinking about it is this. Label the vertices that contain the hypotenuse A and B, and let C be the vertex with the 90 degree angle. As we all should know, the shortest distance between A and B is a straight line (which happens to be the hypotenuse). Now follow A to C and C to B. This is NOT a straight line and hence the length of the hypotenuse is less than the length of the two other sides summed together.

I think the confusion had more to do with trying to understand why, if you can add right-angle vectors A + B = C, you can't determine sides of a right triangle the same way.

 

[AlephZero]

Actually there is nothing wrong with your math, because if you define the distance between two points (x1,y1) and (x2,y2) to be |x2-x1| + |y2-y1|, that "length" definition has many of the same properties as "Euclidean distance" defined as sqrt((x2-x1)^2 + (y2-y1)^2). In some math applications, "length" IS defined that way, because it makes life simpler.

However it is not much use for doing physics, because it makes the length of a line depend on the orientation of the coordinate system that you use to measure it. Experiment shows that the "real world" does not behave that way, and that physics is the same whatever coordinate system you do it in.

For example by your definition of length, a "circle" is actually a square, with the sides oriented in specific directions relative to the X and Y axes. If you rotate your "circle" through an arbitrary angle (say 30 degrees), then either it has to deform into a different shape, or it isn't a circle any more. This is not a useful way to model the behaviour of the "real universe", at least with our current understanding of how the "real universe" behaves.

 

[Shawn Garsed]

I think I was confusing displacement with distance travelled, which are not equal. I would still like some answers to the PS of my original post as I can't find a good website anywhere.

 

[zgozvrm]

I think I was confusing displacement with distance travelled, which are not equal. I would still like some answers to the PS of my original post as I can't find a good website anywhere.

You might try Purplemath:
http://www.purplemath.com/modules/index.htm