Question

Geometry :D
Suppose f and g are two isometries such that f(A) = g(A), f(B) = g(B), and f(C) = g(C) for some nondegenerate triangle ΔABC. Show that f = g

Answer 1

I feel like this should be easy, but still getting used to all the geometry theorems and what not.

Answer 2

@freckles know anything about this stuff?

Answer 3

@ganeshie8 Know any of this stuff?

Answer 4

idk much but i found this
#3
http://www.ms.uky.edu/~droyster/courses/spring04/homework/restricted/Homework2%20Solns.pdf

@ikram002p did non eucledean geometries a couple of years ago

Answer 5

Ah, nice find! I found something else online, but it used a theorem that I couldnt use based on the sequence of my text. But alright, if ikram knows some of this stuff, Ill see what he can do when I have questions. I think I can just use what that link gives. Thanks :)

Answer 6

good luck! :)

Answer 7

hey first u need to know what isometries functions means, it means a rigid transforms function which maps a shape to some where else but without changing size in other way both origin and transforms are congruent.

Answer 8

lets do it in geometry style

Answer 9

so since both are isometries we have 3 congruent triangles, now here is a thing why it called rigid transformation it since u do not change it size and shape , like triangle do not become a circle for example or dont become another triangle with same size but different sides that cannot happen.

Answer 10

now step two, it says if
f(A) = g(A), f(B) = g(B), and f(C) = g(C), then show Show that f = g
in euclidean geometry its ok if u wanna only graph it that works like this it would be a proof by itself

Answer 11

now since you wanna it theoretically

f(A)=g(A)
g^-1o f(A)=g^_1 o g(A) =A
so if u apply the function g^-1 o f on the other two points u got
g^-1 o f(B)=B
g^-1 o f(C)=C
according to a theorem (idk if ts given in ur book or u need to prove it let me know in both cases )
g^-1o f must be the identity which means :-

g^-1 o f(P)=P , for any P on the triangle
g o g^-1 o f(P)= g(P)
f(P)=g(P)
which is done :D
let me know if something is not clear enough, good luck !!