## AonZ Help please :) Yr11 3 unit question on trig In triangle ABC, acosA = bcosB. Prove using the cosine rule, that the triangle is either isosceles or right-angled 10 months ago 10 months ago

1. ajprincess

Do u knw cos rule?

2. AonZ

yep...

3. ajprincess

good:) so vat is cosA?

4. AonZ

b^2 + c^2 - a^2 / 2bc

5. ajprincess

cosB=?

6. AonZ

a^2 +c^2 - b^2 / 2ac

7. ajprincess

good:)

8. ajprincess

Nw if u r to prove that the triangle is isoscles u need to show that a=b or b=c or a=c. if u r to show it is a right angled triangle u should show pythagoras theorem is valid. $a\cos A=b\cos B$ $a* \frac{ b^2+c^2-a^2 }{ 2bc}=b*\frac{ a^2+c^2-b^2 }{ 2ac }$ vat do u get when u simplify this? @AonZ

9. AonZ

2c^3 a^2 - 2a^4 c - 2c^3 b^2 + 2b^4 c = 0

10. ajprincess

u can factor out 2c.

11. AonZ

2c ( b^4 - c^2 b^2 + a^2 c^2 - a^4) = 0 this right so far?

12. ajprincess

yup:) good:) nw divide by 2c both sides.

13. AonZ

ok done that. btw im writing it down

14. ajprincess

ok:)

15. ajprincess

u will get this. b^4 - c^2 b^2 + a^2 c^2 - a^4=0 try to factorise this.

16. AonZ

(b^2 - a^2)(b^2 +a^2 -c^2)

17. AonZ

(b+a)(b-a)(b^2+a^2 - c^2)

18. ajprincess

(b+a)(b-a)(b^2+a^2 - c^2)=0 Divide by (b+a)(b-a) both sides. vat do u get?

19. AonZ

b^2 +a^2 - c^2 = 0

20. AonZ

you can tell its right angle by looking at this.. but how do you tell if its isosceles?

21. ajprincess

u r nt asked to prove both right? either one of that?

22. AonZ

oh i guess :)

23. AonZ

i still think it means to prove that the triangle must be right or isoscles but it cant be scalene

24. AonZ

but can we show it isoscles anyways?

25. ajprincess

let me think. hav to go nw. will be back in 15 mnts

26. gohangoku58

(b+a)(b-a)(b^2+a^2 - c^2) =0 gives (b+a)=0 OR (b-a)= 0 OR (b^2+a^2 - c^2)=0 so, b-a = 0 --->a=b----->isosceles (b^2+a^2 - c^2)=0 --->right as you already know.

27. AonZ

ohh xDDDD THANKS!! Shame i can only give medal to 1 person :(

28. gohangoku58

hey, i am not here for medals ;)

29. ajprincess

Welcome:) you can give it to @gohangoku58

30. ajprincess

cos he/she deserves it most:)

31. gohangoku58

:O You made him solve the whole thing and i deserve it!??

32. AonZ

lol

33. ajprincess

yup:) it was u who pointed out the isoscles one:)