Question

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

Answer 1

Do u knw cos rule?

Answer 2

yep...

Answer 3

good:) so vat is cosA?

Answer 4

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

Answer 5

cosB=?

Answer 6

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

Answer 7

good:)

Answer 8

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

Answer 9

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

Answer 10

u can factor out 2c.

Answer 11

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

Answer 12

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

Answer 13

ok done that. btw im writing it down

Answer 14

ok:)

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

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

Answer 19

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

Answer 20

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

Answer 21

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

Answer 22

oh i guess :)

Answer 23

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

Answer 24

but can we show it isoscles anyways?

Answer 25

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

Answer 26

(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.

Answer 27

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

Answer 28

hey, i am not here for medals ;)

Answer 29

Welcome:) you can give it to @gohangoku58

Answer 30

cos he/she deserves it most:)

Answer 31

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

Answer 32

lol

Answer 33

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