Home
Class 12
MATHS
In a scalene triangle A B C ,D is a poin...

In a scalene triangle `A B C ,D` is a point on the side `A B` such that `C D^2=A D D B ,` sin `s in A S in B=sin^2C/2` then prove that CD is internal bisector of `/_Cdot`

Text Solution

Verified by Experts

Let `angle ACD = alpha`
`rArr angle DCB = (C - alpha)`

Applying the sine rule of `Delta ACD and " in " Delta DCB` respectively, we
`(AD)/(sin alpha) = (CD)/(sin A) and (BD)/(sin (C - alpha)) = (CD)/(sin B)`
`rArr (AD.BD)/(sin alpha. sin (C - alpha)) = (CD^(2))/(sin A. sin B)`
`rArr (1)/(2) [cos (2 alpha - C) - cos C] = sin^(2).(C)/(2)`
`rArr (1)/(2) [cos (2alpha - C) -1 + 2 sin^(2).(C)/(2)] = sin^(2).(C)/(2)`
`rArr cos (2 alpha - C) = 1`
`rArr alpha = (C)/(2)`
Thus, CD is the internal angle bisector of anlge 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

A B C is a triangle. D is a point on A B such that D=1/4A B\ a n d\ E is a point on A C such that A E=1/4A Cdot Prove that D E=1/4B C

A B C is a triangle. D is a point on A B such that A D=1/4A B and E is a point on A C such that A E=1/4A Cdot Prove that D E=1/4B Cdot

In figure D is a point on side BC of a DeltaA B C such that (B D)/(C D)=(A B)/(A C) . Prove that AD is the bisector of /_B A C .

In Figure, A B C is a triangle in which /_B=2/_C . D is a point on side B C such that A D bisects /_B A C a n d A B=C DdotB E is the bisector of /_B . The measure of /_B A C is 72^0 (b) 73^0 (c) 74^0 (d) 96^0

In A B C , A D is a median. Prove that A B^2+A C^2=2\ A D^2+2\ D C^2 .

In a triangle ABC, D is a point on BC such that AD is the internal bisector of angle A . Let angle B = 2 angle C and CD = AB. Then angle A is

In a quadrilateral A B C D , given that /_A+/_D=90o . Prove that A C^2+B D^2=A D^2+B C^2 .

A B C D is a parallelogram. P is a point on A D such that A P=1/3\ A D\ a n d\ Q is a point on B C such that: C Q=1/3B Cdot Prove that A Q C P is a parallelogram.

A B C D is a parallelogram. E is a point on B A such that B E=2\ E A\ a n d\ F is a point on D C such that D F=2\ F Cdot Prove that A E\ C F is a parallelogram whose area is one third of the area of parallelogram A B\ C Ddot

A point D is taken on the side B C of a A B C such that B D=2d Cdot Prove that a r( A B D)=2a r( A D C)dot