Question

There are many different proofs of the Pythagorean Theorem. Research and choose three different proofs. Outline and provide an explanation foreach

Answer 1

@Preetha

Answer 2

@Hero

Answer 3

@jes230

Answer 4

A. All three sides. Also known as the SSS Postulate.



In this first case we have selected the three sides (SSS) from each block and put them together. The red "hatch" marks show which sides correspond and are congruent. The triangles are complete and congruent without the angle parts. However, it should be noted that the angles in the completed triangles are congruent. Knowing that the three corresponding sides are congruent is enough to guarantee that all the parts of the triangles are congruent.

B. 1st side, angle between sides, 2nd side. Also known as the SAS Postulate.



We have selected two sides and the angle formed by those two sides (SAS) as our building parts, but this time the triangles are not complete. Notice how congruent angles have to be marked by drawing little arcs and putting the hatch marks on them. If the triangles could be completed with the missing parts, would they be guaranteed to be congruent? Absolutely! Drawing two sides and the angle between them is enough to guarantee that the triangles are congruent and all the parts are congruent.

C. 1st angle, side between angles, 2nd angle. Also known as the ASA Postulate.



This time two angles and the side connecting them (ASA) were taken and put together in the workspaces. By imagining the finished triangles with the unused parts, we again would find that the triangles are guaranteed to be congruent. Drawing two corresponding angles and the side between them is enough to guarantee that the triangles are completely congruent with all parts congruent.

D. 1st angle, 2nd angle, side not between them. Also known as the AAS Theorem.

Note: This case can be formally proved. That's why it is called a theorem instead of a postulate, which is accepted as true without proof. Geometry has far more theorems than it does postulates.



This is the strangest of all the figures we have made because it doesn't even connect up as the other drawings have done. Nevertheless, if you completed the figure with your leftover parts, can you see that the triangles would be guaranteed to be congruent to each other? As in all the other cases, all the corresponding parts of the triangles are congruent.

Summary: There are only four groups of parts which you can use to make two congruent triangles: SSS, SAS, ASA, and AAS. There are no other combinations of parts which can guarantee congruent triangles.

Answer 5

i just need three simple proofs i will do the rest. i researched and cant find anything at al

Answer 6

are those part of the pythagorean theorem though?

Answer 7

angle side angle ASA then there is side,side,side, SSS then side angle side SAS

Answer 8

does that help?

Answer 9

alright i didnt know if they were the proofs or not. i thought the proofs were formulas.

Answer 10

like the actual theorem: a^2+b^2=c^2

Answer 11

oh i am so sorry thats not what you were looking for my bad, those are proofs but not for what YOU are doing Im sorry

Answer 12

its ok lol i was getting confused. @hero do you know the proofs?

Answer 13

oh nvm Euclid's proof is one I know

Answer 14

do you know what the actual proof is though?

Answer 15

look it up in wikipedia - outline, here is how the proof in Euclid's Elements proceeds. The large square is divided into a left and right rectangle. A triangle is constructed that has half the area of the left rectangle. Then another triangle is constructed that has half the area of the square on the left-most side. These two triangles are shown to be congruent, proving this square has the same area as the left rectangle. This argument is followed by a similar version for the right rectangle and the remaining square. Putting the two rectangles together to reform the square on the hypotenuse, its area is the same as the sum of the area of the other two squares

Answer 16

there is a few more but I remember try to do research on this proof in peticular

Answer 17

i will use that one and two more

Answer 18

do either of you have an idea of how i can explain them?

Answer 19

there is proof by rearangment too and proof by differentials

Answer 20

and algabraic proofs

Answer 21

alright ill try to figure it out.

Answer 22

not really, I cant really explain them because i don t want to give you the wrong idea

Answer 23

i dont really understand them so anything will help

Answer 24

thanks do you know how to explain them?

Answer 25

yes i know but i dont know how to explain them