## aroub 3 years ago Given a triangle ABC such that B and C are fixed. Point A varies such that segment AB is congruent to segment AC. Find the locus of A. Umm, an angle bisector maybe?

1. AbhimanyuPudi

perpendicular bisector of BC

2. aroub

Aoh -.- I get it now! Because they said that the sides are congruent so they are equidistant so it's perpendicular bisector!! Let me draw it and tell me if it's right or wrong, okay? :)

3. AbhimanyuPudi

sure

4. klimenkov

Sure. You are welcome.

5. aroub

|dw:1352049851819:dw|

6. hartnn

do u know how to solve this algebraically ?

7. hartnn

how to get to 'perpendicular bisector of BC'

8. aroub

Emmm, i don't think so.. like what do you mean? show me? :)

9. AbhimanyuPudi

|dw:1352049933632:dw|

10. aroub

Does it matter?

11. AbhimanyuPudi

Yea..the point A can be above BC or below BC according to that figure

12. aroub

Okay, thank youu!

13. hartnn

ok, i choose point B as origin of my co-ordinate axes and point C on x-axis. as they are fixed i can put them wherever i want. now let co-ordinates of A be (x,y) we need a locus in terms of x and y let me draw it. |dw:1352050085145:dw| now AB=AC will give you, using distance formula, \(\huge x^2+y^2=(x-a)^2+y^2\) can u simplify this and tell me what u get ?

14. AbhimanyuPudi

welcome :-)

15. klimenkov

|dw:1352050205624:dw|

16. aroub

Oh, oh! No.. not yet at least. Let me know how to find the locus first :P Thank you though!!

17. hartnn

if u simplify that u get x=a/2 which is a line that bisects BC , hence 'bisector' and is perpendicular to BC (as it is vertical line, its perpendicular to x-axis) hence, 'perpendicular' did u get my explanation ?

18. AbhimanyuPudi

A small correction @hartnn It is \[x ^{2}+y ^{2} = x ^{2} + (y-a)^{2}\]

19. AbhimanyuPudi

Ok got it..that isn't the correction..the correction is B=(a,0)

20. hartnn

correction is in diagram! not there |dw:1352050580220:dw|

21. AbhimanyuPudi

Ya ya..that's what I was talking about

22. aroub

Yes I did! So you can get it with solving.. Not by memorizing the conditions or cases. Nice!

23. hartnn

always!