Question

I will fan and medal !!!
Prove that the two circles shown below are similar.

Answer 1

@pooja195 Please, please help I have been stuck with this for hours.

Answer 2

It easily follows from the mathematical definition of being "similar". What is that defintion?

Answer 3

Of similar?

Answer 4

What does it mean that two circles are similar?

Answer 5

It means that they are both equal to the fixed point

Answer 6

No that is that is not the definition of two objects being similar: http://www.mathsisfun.com/geometry/similar.html

Answer 7

Oh okay, so what do I put to proof that the circles are similar?

Answer 8

Two objects are similar if you can resize, rotate, reflect and translate rotate one of the objects into the other

Answer 9

Yep, thank you, but how do I prove it using those two circles?

Answer 10

So you have to prove that, after a combination of resizing, rotation, reflection and translation, you can place one circle directly on top of the other. Does that make sense?

Answer 11

Yes, but I don't know how to prove it. D:

Answer 12

What I mean is you explained it but with the coordinates of the circle I don't know how to prove.

Answer 13

So, let's try and place the little circle directly over the bigger circle. In order to do that, i need the little circle the same size and centered exactly where the bigger circle is. That would prove that they're similar

Answer 14

Oh okay, so how do I make them the same size and centered exactly?

Answer 15

So it basically comes down to the question of: How many units do I need to shift the centre of the little circle upwards and leftwards to get it to be at the centre of the bigger circle? And then how much do i need to scale the little circle by to make it the same size as the bigger circle?

Answer 16

sorry, that should say rightwards and not leftwards

Answer 17

You need to dilate it.

Answer 18

To know how much you need to scale the little circle, consider the radius of the little circle and the radius of the bigger circle

Answer 19

Oh okay.

Answer 20

So its (3,4) (0,-1)

Answer 21

ummm, yes, you need to move the centre from (0,-1) to (3,4) so how many units do you need to shift it from (0,-1) to (3,4)?

Answer 22

*how many units rightwards and upwards do you need to shift it from (0,-1) to (3,4)?

Answer 23

(3,5)

Answer 24

:)

Answer 25

So in proper terminology, we say that we are applying a translation of 3 units rightwards and 5 units upwards

Answer 26

Okay. Then what do we do (By the way, thanks so much I'm actually understanding)

Answer 27

If you would like to get a bit more technical, then we are applying a translation vector \[\left(\begin{matrix}3 \\ 5\end{matrix}\right)\]. Tells you about it here: http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/transformationsrev2.shtml

Answer 28

Okay, so now we have translated it and have the little circle's centre right where the bigger circle's centre is, we need to scale the little circle up so that we can get the little circle directly over the bigger circle

Answer 29

Okay

Answer 30

Following me? :P Any idea on how to do it?

Answer 31

Yep but I just don't know how to do it next

Answer 32

okay, so what is the radius of the little circle and what is the radius of the bigger circle?

Answer 33

Um by let me see.

Answer 34

By 2pir

Answer 35

sorry, i don't understand :P what is 2pir?

Answer 36

Sorry I ment x=r2/r1

Answer 37

Yup! :D

Answer 38

Actually, i've just realised a slight complication :P

Answer 39

So how do I plug it in D:

Answer 40

oh... let's deal with that first

Answer 41

so do you know what is r2 and r1?

Answer 42

Radius 1 and Radius 2?

Answer 43

And what are those values?

Answer 44

I don't know :/

Answer 45

Radius 1 is the radius of the little circle, and radius 2 is the radius of the bigger circle. So what are the radii of the little and bigger circles?

Answer 46

I don't know. ://

Answer 47

Can you tell me but show me how you got it so I learned

Answer 48

Radius 1 is the radius of the little circle, and radius 2 is the radius of the bigger circle. So what are the radii of the little and bigger circles?

Answer 49

How do I get the redii

Answer 50

The length of the red line is the length of radius 1 and length of green line is the length of radius 2

Answer 51

How do I calculate it though?

Answer 52

So, the endpoints of the green line are (-2, 4) and (3,4) so the length of the green line is?

Answer 53

And the length of the green line IS radius 2

Answer 54

(5, 0)

Answer 55

Well, the length of the green line is 5

Answer 56

THE LENGTH IS 2 FOR THE RED LINE

Answer 57

:D

Answer 58

So then what ://

Answer 59

So r1=2, r2=5 and then in order to scale the little circle to the big circle, we need ti scale it by a scale factor of r2/r1

Answer 60

it equals to 2.5

Answer 61

Indeed :)

Answer 62

So whats after that

Answer 63

Lol, so what have u done so far?

Answer 64

I have

Answer 65

First you have to apply a translation of 3 units rightwards and 5 units upwards. Which is a translation vector.
Then you use x=r2/r1
which the bigger circle equals (5,0)
and the smaller equals (2,0)
this equals to 2.5.

Answer 66

Hmmm, so the translation thing is nice and clear

Answer 67

The rest isn't as clear :P

Answer 68

How do I fix it, may you help :(

Answer 69

For the second part we wish to say this:
We apply a dilation of scale factor ___ about the centre (__,__) to the smaller circle

Answer 70

You need to fill in the blanks :P

Answer 71

Does that make sense?

Answer 72

Okay and the blank space for the first one is x=r2/r1 and the second blank space is 2.5?

Answer 73

No, x=r2/r2=5/2=2.5

Answer 74

x IS the scale factor

Answer 75

the scale factor IS 2.5 which you calculated by dividing r2 by r1

Answer 76

for the first one I put r2/r1 and for the second I put 5/2=2.5

Answer 77

nono, they are both the same thing haha

Answer 78

Oh yesss so for the first I put 5/2= 2.5

Answer 79

yes :)

Answer 80

And for the second I put (2,5)

Answer 81

no, not (2,5)

Answer 82

it should be (3,4) because that is where the centre of little circle is.

Answer 83

Ohh okay, so I have
First you have to apply a translation of 3 units rightwards and 5 units upwards. Which is a translation vector.
I then apply a dilation of the scale factor 5/2=2.5 about the center (3,4) to the smaller circle.

Answer 84

You don't need to mention about the translation vector thing. Don't worry about it.

Answer 85

Apart from that, yes! :D

Answer 86

That's the proof

Answer 87

YAYAYAYA. so then I put at the end "Which proves that the two circles will be similar"?

Answer 88

Yes, because as we have proved, after some combination of dilation, rotation, translation and reflection, we can make one circle completely identical to the other. ^_^

Answer 89

Oh okay thank you so much!!!!

Answer 90

np ^_^

Answer 91

.