## aroub 3 years ago Consider the figure below. Given that the two concentric circles are fixed. Find the locus of midpoint MN as M traces on outer circle, keeping O,N and M are collinear.

1. aroub

|dw:1352052268576:dw|

2. aroub

Just a line?

3. aroub

@phi ? :)

4. cwrw238

the locus is a circle - can you say what its radius is?

5. aroub

Yes, I can. But only if you told me how did you get it and which circle?

6. phi

keeping O,N and M are collinear. means a fixed length (like a stick)

7. phi

if M traces out the outer circle, what does the rest of the "stick" do?

8. aroub

Another circle? With radius OM?

9. cwrw238

no the radius is ON + (MN/2) - do you see why?

10. aroub

The other circle that will be formed is it like this? |dw:1352053569404:dw|

11. aroub

Great :D So how did you get the radius? It's OM + MN?

12. cwrw238

no!!! sorry - the circle rund in between the 2 original circles

13. phi

The mid point of MN is between points M and N . As the radius OM traces out a circle, what does that mid point do?

14. aroub

But if you thought of it, they said that M traces on the outer circle keeping O,N and M collinear so it forms a circle outside?!

15. aroub

Aooh, wait! phi's answer makes more sense :P So it makes two other circles or just the circle between?

16. phi

you put a dab of paint on the "stick" at the point halfway between M and N where is the "smear" as you turn the stick?

17. phi

|dw:1352054278598:dw|

18. aroub

The "smear" will be between the two circles forming a circle, right?

19. phi

yes the radius is the "average" of ON and OM or (ON + OM)/2 which is the same as ON + NM/2

20. aroub

This was my last question in locus ( I think and I really hope so ) You're amazing, I'm not kidding :P I love how you think of locus! Like in a very simple way and understandable. Thank you soooooooo mucchhh!! :D