# A little geometry

juef
Hi all,

I'm having a hard time proving something. Let's say you have a triangle with a 90° angle. How can I prove that the length of the segment between the middle of the hypotenuse and the 90° angle is half the length of the hypotenuse itself?

Thanks a lot, and sorry for my pityful english! :)

john

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Homework Helper
Uh... isn't that what the picture says anyways? Why do you need to prove it if the picture is telling you it... or do you need to proce it for all cases?

juef
Well, I have to prove that if one of the two (the 90° angle or the length that is half the length of the hypotenuse) is true, then the other is too.

metacristi
juef said:
Hi all,

I'm having a hard time proving something. Let's say you have a triangle with a 90° angle. How can I prove that the length of the segment between the middle of the hypotenuse and the 90° angle is half the length of the hypotenuse itself?

Thanks a lot, and sorry for my pityful english! :)

john

I do not know your level of knowledge in geometry,the simplest solution I see at first sight is to use Stewart's relation (knowing also that $$b^2+c^2=a^2$$ and that the hypothenuse is split into two equal segments a/2).

See http://mathworld.wolfram.com/StewartsTheorem.html

With the notations used there we have:

[PA3]*[A1A2]2+[PA2]*[A1A3]2=[PA1]2*[A2A3]+[PA2]*[PA3]*[A2A3]

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Staff Emeritus
Gold Member
I think Stewart's theorem is an overkill in this case.

At the middle point of the hypothenuse, draw a segment parallel to one of the other two sides, and look for congruent triangles.

Homework Helper
Gold Member
Try completing the rectangle.

Staff Emeritus
Gold Member
How about using the (converse of the) fact that the angle in a semicircle is always a right angle ?

<Not sure if you've done circles yet. If not, I defer to robphy's suggestion.>

metacristi
ahrkron said:
I think Stewart's theorem is an overkill in this case.

Well depends on the knowledge level.After all Stewart's relation can be easily deduced by applying Pitagoras' generalized theorem two times,rearranging a bit the equations and taking also into account that cosX+cos[π-X]=0.

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juef