Home
Class 12
MATHS
If in a triangle A B C ,/C=60^0, then pr...

If in a triangle `A B C ,/_C=60^0,` then prove that `1/(a+c)+1/(b+c)=3/(a+b+c)` .

Text Solution

Verified by Experts

By the cosine formula, we have
`c^(2) = a^(2) + b^(2) - 2ab cos C`
or `c^(2) = a^(2) + b^(2) - 2ab cos 60^(@) = a^(2) + b^(2) - ab` (i)
Now `(1)/(a + c) + (1)/(b + c) - (3)/(a + b + c)`
`=[((b + c) (a + b +c) + (a + c) (a + b+ c) -3 (a + c) (b + c))/((a + b) (b + c) (a + b + c))]`
`= ((a^(2) + b^(2) - ab) - c^(2))/((a + b) (b + c) (a + b+ c)) = 0` [from Eq. (i)]
or `(1)/(a + c) + (1)/(b + c) = (3)/(a + b + c)`
Promotional Banner

Topper's Solved these Questions

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE|Exercise Exercise 5.1|12 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE|Exercise Exercise 5.2|8 Videos

  • PROGRESSION AND SERIES

    CENGAGE|Exercise ARCHIVES (MATRIX MATCH TYPE )|1 Videos

  • PROPERTIES OF TRIANGLE, HEIGHT AND DISTANCE

    CENGAGE|Exercise Question Bank|32 Videos

Similar Questions

Explore conceptually related problems

If in a triangle ABC,/_C=60^(@), then prove that (1)/(a+c)+(1)/(b+c)=(3)/(a+b+c)

If in a triangle ABC,/_C=60^(@), then prove that (1)/(a+c)+(1)/(b+c)=(3)/(a+b+c).

In a triangle ABC,/_C=60^(@), then prove that: (1)/(a+c)+(1)/(b+c)=(3)/(a+b+c).

If C=60^(@). then prove that (1)/(a+c)+(1)/(b+c)=(3)/(a+b+c)

If in a triangle ABC, angleC=60^(@) , then (1)/(a+c)+(1)/(b+c)-(3)/(a+b+c) =

If in triangle ABC, C = 90^circ then prove that tan^(-1) (a/(c+b)) + tan^(-1) (b/(a+c)) = pi/4 .

If, in a triangleABC, A=60^(@) , then: (1+a/c + b/c)(1+c/b - a/b)=

In a triangle ABC, prove that (1+(b-c)/a)^a*(1+(c-a)/b)^b*(1+(a-b)/c)^cle1 .

In a triangle ABC prove that a/(a+c)+b/(c+a)+c/(a+b)<2

In any triangle ABC,prove that a sin (B-C)+b sin(C-A)+c sin(A-B)=0