Question

the lengths of the sides of a triangle are b and c let S be the length of the angle bisector of the angel between the two given sides . the length of the third side of the triangle is​

Answer 1

Given :    The lengths of the sides of a triangle are b and c let S be the length of the angle bisector of the angel between the two given sides .

To Find : the length of the third side of the triangle

Solution:

Assume that angle A is 2α

Area of  ΔABC = (1/2) bcsin2α

Area of ΔADC = (1/2) bSsinα

Area of ΔADB = (1/2) cSsinα

Area of  ΔABC = Area of ΔADC + Area of ΔADB

=> (1/2) bcsin2α = (1/2) bSsinα + (1/2) cSsinα

=> bcsin2α =  (b + c) Ssinα

=> bc2sinαcosα =  (b + c) Ssinα

=> cosα =  (b + c) S /2bc

cos2α = 2cos²α  - 1

=> cos2α = 2  {(b + c) S /2bc}²  - 1

also cos2α =  (b² + c² - a²)/2bc     Cosine Rule :

Equating Both :

(b² + c² - a²)/2bc  = 2 {(b + c) S /2bc}²  - 1

Simplify to find a in terms of b. c and S

or substitute values of b , c and S if given to find a ( this side of triangle )

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Answer 2

Answer:

$\sqrt{(1-\frac{s^2}{bc})(b+c)^2}$

Step-by-step explanation:

$\sqrt{(1-\frac{s^2}{bc})(b+c)^2}$