Home
Class 12
MATHS
Prove that ("a c o s"A+b cosB+ccosC)/(a+...

Prove that `("a c o s"A+b cosB+ccosC)/(a+b+c)=r/Rdot`

Text Solution

Verified by Experts

We have
`a cos A + b cos B + c cos C`
`= R(2 sin A cos A + 2 sin B cos B + 2 sin A sin C)`
`= R (sin 2A + sin 2B + sin 2C)`
`= 4R sin A sin B sin C`
and `a + b + c = 2R (sin A + sin B + sin C)`
`= 8 R cos (A//2) cos (B//2) cos (C//2)`
`rArr (a cos A + b cos B + c cos C)/(a + b + c)`
`= (4R sin A sin B sin C)/(8R cos A //2 cos B//2 cos C//2)`
`= ((2 sin.(A)/(2) cos.(A)/(2))(2sin .(B)/(2) cos.(B)/(2)) (2sin.(C)/(2) cos.(C)/(2)))/(2cos.(A)/(2) cos.(B)/(2) cos.(C)/(2))`
`= 4 sin.(A)/(2)sin.(B)/(2) sin.(C)/(2) = (r)/(R)`
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

Prove that (a cos A+b cos B+cs C)/(a+b+c)=(r)/(R)

Prove that (acos A+b cos B+c cos C)/(a+b+c)=(r)/(R)

Prove that a cos A+b cos B+c cos C<=s

Prove that in a triangle ABC, (cosB-cosC)/(cosA+1) =(c-b)/a

If A+B+C = pi , prove that : cosA- cosB - cosC = 1-4sinA//2cosB//2cosC//2 .

If A+B+C=180^0 , prove that : cos^2 A + cos^2 B + cos^2 C + 2cosA cosB cosC=1 .

Prove that a cos A+b cos B+c cos C=4R sin A sin B sin C

Prove that: ("sin"(A-B))/(cos A cosB)+(sin(B-C))/(cos B cos C)+("sin"(C-A))/(cos C cos A)=0

In DeltaABC , prove that: (c-bcosA)/(b-ccosA)=(cosB)/(cosC)

If A+B+C=pi , prove that : cosA + cosB-cosC=4cos(A/2) cos(B/2) sin(C/2) -1