Cyclic?

Geometry Level 3

Let $\tau_{A, B}$ represent a linear translation in the $xy$ -plane from point $A$ to point $B$ .

Given the points

$A=(21,42)$
$B=(32,43)$
$C=(-12,6)$
$D=(-112,206),$

if $\tau_{A, B}\tau_{B,C}\tau_{C,D}\tau_{D,A}(2,8)=(a,b)$ , what is $a+b?$

The answer is 10.

1 solution

Geoff Pilling
Jan 25, 2017

This is a cyclic translation.

$\tau_{A, B}$ takes you from point $A \rightarrow B$

$\tau_{B, C}$ takes you from point $B \rightarrow C$

$\tau_{C, D}$ takes you from point $C \rightarrow D$

$\tau_{D, A}$ takes you from point $D \rightarrow A$

So, the net result of all these translation is no translation at all.

So, you will get back to point $(a,b) = (2,8)$

$2 + 8 = \boxed{10}$

Before I read your wiki I did the translations from right to left and still ended up back where I started. Is it the case that any rearrangement of a series of transformations will yield identical results, or does this only work if the transformations can be arranged in cyclic order?

Brian Charlesworth - 4 years, 4 months ago

Interesting... I never thought about that, but I suppose you are right... Like adding vectors, I think it doesn't matter too much about order.

Geoff Pilling - 4 years, 4 months ago

Yes, essentially transformations are vectors, I guess, so if they are cyclic the order doesn't matter. If they aren't cyclic then for $n$ transformations you could end up in a maximum of $n!$ different locations. "Lost in Translation" .... :p

Brian Charlesworth - 4 years, 4 months ago

@Brian Charlesworth – But wait a sec... For n = 2 you can only end up in 1 location?

Geoff Pilling - 4 years, 4 months ago

@Geoff Pilling – Erps. I can't draw accurately, apparently. Yeah, summation of vectors is order invariant, so cyclic or not, transformations will be as well.

Brian Charlesworth - 4 years, 4 months ago

Cyclic?

The answer is 10.

1 solution

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