Home
Class 12
MATHS
If the incircle of the triangle ABC pass...

If the incircle of the triangle ABC passes through its circumcenter, then find the value of `4 sin.(A)/(2) sin.(B)/(2) sin.(C)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
`sqrt2 -1`
Distance between circumcenter (O) and incenter (I) is `sqrt(R^(2) -2r R)`
If incircle of triangle ABC passes through it's circumcenter,
`sqrt(R^(2) - 2r R) = r`
`rArr ((r)/(R))^(2) + 2((r)/(R)) - 1 = 0`
`rArr (r)/(R) = (-2 +- sqrt(4 + 4))/(2)`
`rArr (r)/(R) = sqrt2 -1` (as `(r)/(R) gt 0`)
`rArr 1 + (r)/(R) = sqrt2`
`rArr 4 sin.(A)/(2) sin.(B)/(2) sin.(C)/(2) = sqrt2 -1`
Promotional Banner

Topper's Solved these Questions

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Concept application exercise 5.9|5 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Concept application exercise 5.10|8 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Concept application exercise 5.7|4 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise ARCHIVES (NUMERICAL VALUE TYPE )|8 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE PUBLICATION|Exercise All Questions|1119 Videos

Similar Questions

Explore conceptually related problems

Find the value of 2ac sin ((A-B+C)/2) for /_\ABC

Find the value of 2ac sin((A-B+C)/(2))" for "DeltaABC .

If in triangle A B C ,/_C=45^0 then find the range of the values of sin^2A+sin^2Bdot

A triangle ABC is inscirbed in a circle. If half the sum of the square of its three sides is equal to twice the square of the diameter of the circle, then the value of sin^(2)A + sin^(2)B + sin^(2)C will be-

If cos(A+B+C)=cosAcosBcosC , then find the value of (8sin(B+C)sin(C+A)sin(A+B))/(sin2Asin2Bsin2C)

If A+C=2b, then the value of (cos C - cos A)/(sin A-sin C) is-

ABC is a right angled triangle, then the value of sin^(2)A + sin^(2)B + sin^(2)C will be-

In triangle ABC, if a^(2) + b^(2) +c^(2) -bc -ca -ab=0, then the value of sin ^(2)A+ sin ^(2)B+sin ^(2)C is -

The perimeter of a triangle ABC iws equal to 2 (sin A+sin B+ sin C). If a=1, then the value of angle A is-

In any triangle ABC, the value of (b^(2)sin 2C + c^(2) sin 2B) is-