Question

Triangle dilation question

Answer 1

well i just learned something today. so let me explain it

Answer 2

take a point any point POINT P? thats a good one

Answer 3

it will be 2.5 times further from the 'dilation' when you scale it by 2.5

Answer 4

i know the picture is kinda bad but do you see that there are two lines

Answer 5

the really short line is the distance from P to the dilation

Answer 6

the really long line is where "p is going" when we "scale" our triangle

Answer 7

do you get that part?

Answer 8

If I call (x',y') the new coordinates of P, we can write:
\[(x',y')=2,5*(1,-2)\]

Answer 9

namely, each old coordinate of P must to be multiplied by the factor 2.5

Answer 10

lol i guess the dilation is at the origin so yeah its just simple multiplication

Answer 11

im hoping to first show you what DILATION does

Answer 12

@lily1021 by definition, a dilation changes the old coordinates, namely (x,y), into a new coordinates, namely (x,y), those two couple of corrdinates are related each other, by the subsequent relationship:
\[(x',y')=k*(x,y)\]
where k is the dilation coefficient, or the scale dactor, it 's only a definition.

Answer 13

@lily1021 sorry the new coordinates, namely (x',y')....

Answer 14

but michele... just multiplying by k only works if the dilation is at 0,0

Answer 15

let me ask you say we took the same triangle and the dilation was at 256,786 what would change in your equation

Answer 16

yes...... but im concerned that you don't know how you got there

Answer 17

@camper in your case, we have not a simple dilation, but a similitude. The relationship for a similitude is:
\[(x',y')=k*(x,y)+(x _{0},y _{0})\]
where \[(x _{0},y _{0})=(256,786)\]

Answer 18

WHOAH is it really that simple?

Answer 19

just add the new dilation coordinates?

Answer 20

@camper that's right!

Answer 21

learning new things. thats why i do open study.

Answer 22

@lily1021 please, tell me your answer

Answer 23

your answer is the right one! I said before, you have to apply the definition, of dilation

Answer 24

Thank you!

Answer 25

hold on there michele

Answer 26

i did some mathemateleicsm

Answer 27

you cannot simply multiply by K and then add the new coordinates

Answer 28

(the dilation coordinates that is)

Answer 29

@camper Why not?

Answer 30

because when you multiply by K you are dilating on 0,0

Answer 31

which is fine for this problem BUT not for (278, blah blah)

Answer 32

it would be (x - x1, y - y1)K + (x1, y1)

Answer 33

@camper4834 I'm not agree, in the first case, the center of dilation is (0,0), in your case the center of dilation is (278,...)

Answer 34

okay lets do this lets take the original problem and instead make the dilation 10,10

Answer 35

using your formula tell me where the new P is

Answer 36

Ok! tell me your center of dilation, please

Answer 37

10,10

Answer 38

and the dilation coefficient, or scale factor, what is?

Answer 39

the same as before 2.5

Answer 40

Ok! the new corrdinates of P, namely (x',y') are:
\[(x',y')=2.5*(x,y)+(10,10)\]
where (x,y) are the old coordinates of P

Answer 41

go ahead tell me what you get

Answer 42

tell me the old coordinates of P, please

Answer 43

the same as before, (1,-2)

Answer 44

Ok! Then we get, for (x',y') the subsequent corrdinates:
(x',y')=(12.5,5)

Answer 45

you are saying that