if cot A/2:cot B/2: cot =4:3:2 then a:b:c=
Answers
Answer:
As we know by using the half angle formula in sides form, The value of cot(
2
A
) =
(s−b)(s−c)
(s)(s−a)
As we know by using the half angle formula in sides form, The value of cot(
2
B
) =
(s−a)(s−c)
(s)(s−b)
As we know by using the half angle formula in sides form, The value of cot(
2
C
) =
(s−a)(s−b)
(s)(s−c)
Since we know that, cot(
2
A
) , cot(
2
B
) and cot(
2
C
) are in ratio, so if we multiply each of them with △, the ratio remains the same.
So, △cot(
2
A
) , △cot(
2
B
) and △cot(
2
C
) are in ratio
△cot(
2
A
) = s(s−a)
Similarly, △cot(
2
B
) = s(s−b)
Similarly, △cot(
2
C
) = s(s−c)
Dividing each term by s, then also the ratio remains the same, so (s−a) , (s−b) and (s−c) are in ratio of 3:7:9
As we know s=
2
a+b+c
So putting value of s and replacing ration by constant k, we get
b+c−a=6k
a+c−b=14k
a+b−c=18k
Adding equation (1) and equation (2), we get
2c=20k
∴ c=10k
Similarly, adding equation (2) and equation (3), we get
2a=32k
∴ a=16k
Similarly, adding equation (1) and equation (3), we get
2b=24k
∴ b=12k
So, a,b and c are in the ratio of 16:12:10, on diving the ratio by 2 we get, 8:6:5.