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

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Discussion Overview

The discussion centers around the question of why the length of the hypotenuse in a right triangle is not equal to the sum of the lengths of the adjacent and opposite sides. Participants explore this concept from both mathematical and philosophical perspectives, touching on geometry, vector addition, and the nature of distance and displacement.

Discussion Character

  • Philosophical
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about why the hypotenuse is not the sum of the other two sides, suggesting a conceptual link between the hypotenuse and the adjacent and opposite sides as components of displacement.
  • Another participant clarifies that in geometry, the Pythagorean Theorem applies, stating that the hypotenuse is related to the legs of the triangle through the equation C² = A² + B², while also discussing vector addition in physics.
  • Some participants argue about the correctness of vector addition, with one stating that the addition of right-angle vectors is being misunderstood.
  • A participant proposes that the hypotenuse represents the shortest distance between two points, contrasting it with the longer path taken along the triangle's legs.
  • Another participant discusses the implications of defining distance in different ways, noting that certain definitions may not be useful in physics due to their dependence on coordinate systems.
  • One participant acknowledges confusion between displacement and distance traveled, indicating a need for further clarification.
  • Several participants seek resources for algebra exercises, indicating a desire to improve their mathematical skills.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between the hypotenuse and the other sides of the triangle, with no consensus reached on the underlying reasons for the observed mathematical properties. The discussion remains unresolved regarding the conceptual understanding of distance and displacement.

Contextual Notes

Participants highlight the distinction between geometric and physical interpretations of distance, as well as the implications of different definitions of length. There are unresolved questions about the nature of vector addition and its application to the problem at hand.

Who May Find This Useful

Individuals interested in the philosophical aspects of mathematics, geometry, vector physics, and those seeking to improve their algebra skills may find this discussion relevant.

Shawn Garsed
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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.
 
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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.
 
Even if you think of this as "physics" you are adding vectors incorrectly.
 
HallsofIvy said:
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?
 
Shawn Garsed said:
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.
 
gb7nash said:
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.
 
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.
 
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.
 
Shawn Garsed said:
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
 

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