SOLUTION: ABC is an right angled triangle,right angled at B.Internal bisector of angle B meet AC at D.AB=3cm,BC=4 cm, find AD.

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Question 468892: ABC is an right angled triangle,right angled at B.Internal bisector of angle B meet AC at D.AB=3cm,BC=4 cm, find AD.
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
ABC is an right angled triangle,right angled at B.Internal bisector of angle B meet AC at D.AB=3cm,BC=4 cm, find AD.
 

Looking at the big right triangle ABC, and the
fact that the tangent is OPPOSITE%2F%28ADJACENT%29

tan%28%22%3CA%22%29+=+%28BC%29%2F%28AB%29+=+4%2F3

Use the inverse tangent on a calculator to find ∠A:

∠A = 53.13010235°

We know that ∠ABD is 45° because BD bisects the right angle at B

We find ∠ADB by using the fact that
the sum of the interior angles of ᅀABD is 180°

∠A + ∠ABD + ∠ADB = 180° 

∠ADB = 180° - ∠A - ∠ABD

∠ADB = 180° - 53.13010235° - 45° = 81.86989765°

So we use the law of sines:

%28AD%29%2Fsin%28%22%3CABD%22%29=%28AB%29%2Fsin%28%22%3CADB%22%29

Cross-multiplying:

AD%2Asin%28%22%3CADB%22%29=AB%2Asin%28%22%3CABD%22%29

Divide both sides by sin%28%22%3CADB%22%29:
  
AD=%28AB%2Asin%28%22%3CABD%22%29%29%2Fsin%28%22%3CADB%22%29

AD=%283sin%28%2245%B0%22%29%29%2Fsin%28%2281.86989765%B0%22%29

Work out the right side with a calculator:

AD = 2.142857143cm

Edwin