Question

three side a,b,and c of a triangle ABC satisfy the ratio 6b+c:3b+a:2c-a=4:2:1.
If the area of triangle is 8sqrt{14},find the values of a,b and c

Answer 1

Is this the question ?

Answer 2

yes

Answer 3

Okay just thought I'd make really sure before I make any drawings, calculations, etc.

Answer 4

'kay

Answer 5

A) 6b + c = 8c - 4a
B) 3b + a = 4c -2a
C) 6b + c = 6b + 2a

Changing equation C
c = 2a

Equation B
3b + a = 4*2a -2a
3b = 5a

Equation A
6b + c = 8c -4*(c/2)
6b -7c = -2c
6b = 5c

-2a + c =0
-5a + 3b=0
6b -5c =0

I cannot solve these equations. Perhaps I didn't set up these equations properly.

Answer 6

are u familiar with heron's formula for area of a triangle?

Answer 7

Yes - set up a semi-perimeter, etc

Answer 8

right, suppose that sides are x, 2x and 4x then using heron's formula you can evaluate x

Answer 9

okay - good thought

Answer 10

and then set up your equations like this

2c-a=x , 3b+a=2x and 6b+c=4x

Answer 11

Hey I just thought of something - the 3 sides cannot be in the ratio of 4 , 2 and 1
The longest side (4) must be smaller than the sum of the other 2 sides.
It isn;t. ARRGGGHHHHHHH !!!!!!!!!!!!!!!!!!!

Answer 12

good point :-))

Answer 13

x+2x < 4x

Answer 14

I was wondering why I couldn't solve it.
Now I know.

Answer 15

according what you got, @wolf1728
a = a

c = 2a

b = 5a/3

perimeter = a+b+c= a + 5a/3 + 2a = 14a/3
s =1/2 * 14a/3 = 7a/3

area = sqrt(s(s-a)(s-b)(s-c))
8sqrt(14) = sqrt( 7a/3 * (7a/3 - a)(7a/3 - 5a/3)(7a/3 - 2a))
8sqrt(14) = sqrt(7a/3 * 4a/3 * 2a/3 * a/3)
from here, we can solve a first

Answer 16

You still think it is solvable?

Answer 17

oh wait !!!!!!!!! we made a big mistake

Answer 18

yes that works. i got a = 6, b = 10, and c = 12

Answer 19

@RadEn is right, that's not the ratio of sides but a combination of sides :-))

Answer 20

@wolf1728 your work is fine so :-)

Answer 21

RadEn the ratios are supposed to be 4: 2: 1
6 : 10: 12
doesn't work.

Answer 22

oh no, problem is 6b+c:3b+a:2c-a=4:2:1 not a:b:c=4:2:1

Answer 23

okay - that was my problem from the beginning

Answer 24

(6b+c) : (3b+a) : (2c-a) = 4 : 2 : 1

plug in the values of a=6, b=12, and c=12
the ratio be right :)

Answer 25

i mean b=10

Answer 26

for asker @someoneelse , see replies 1, 5 and 15

Answer 27

Hey we ALL got medals!

Answer 28

:-))

Answer 29

Well I think I'll get going. See you later folks.