Answer
425.4k+ views
Hint: Here the properties of bisector are the only information which we use to reach the solution, which are as follows:
* Bisector of an angle A is defined to be the line cutting off A into equal halves which meets the
opposite side. In the below figure, AD is the bisector of angle A. Thus it divides angle A into 2 equal
halves and meets the opposite side BC.
* Bisector of an angle of a triangle divides the opposite side in the ratio of the sides containing the angle.
Implies point D divides BC into the ratio of sides containing angle A which are AC and AB. Thus
$\dfrac{{BD}}{{DC}} = \dfrac{{AC}}{{AB}}$
* Sine rule: Ratios of a side to corresponding opposite angles are equal. $\dfrac{{AB}}{{\sin C}} = \dfrac{{AC}}{{\sin B}} = \dfrac{{BC}}{{\sin A}}$
Complete step-by-step answer:
We start with bisector AD and develop its relation with the interior angles.
Sine rules on the triangle ABD gives the result: $\dfrac{{AD}}{{\sin B}} = \dfrac{{BD}}{{\sin (A/2)}}$
From the property of bisector and sine rule, $BD = \dfrac{c}{{b + c}}a$
$AD = \dfrac{{ca}}{{b + c}}\dfrac{{\sin B}}{{\sin (A/2)}}$
Multiplying numerator and denominator by $2\cos (A/2)$, for simplifying the equation to shorter terms.
$AD = \dfrac{{ca.\sin B.2\cos (A/2)}}{{(b + c)\sin (A/2).2\cos (A/2)}}$
Using the formula $\sin (2\theta ) = 2\sin \theta \cos \theta $ ,
$AD = 2.\dfrac{{ca.\sin B.\cos (A/2)}}{{(b + c)\sin A}}$
Using sine rule: $\dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}}$ we get,
$
AD = 2.\dfrac{{cb\sin B\cos (A/2)}}{{(b + c)\sin B}} \\
= \dfrac{{2cb\cos (A/2)}}{{b + c}} \\
$
From the property of bisector and sine rule,
$AD = \dfrac{{2bc}}{{b + c}}\dfrac{{\cos (A/2)\sin (A/2)}}{{\sin (A/2)}}$
Simplifying,
$AD = \dfrac{{bc}}{{\sin (A/2)}}\dfrac{{\sin A}}{{\sin (A/2)}}$
Using $\sin A = \dfrac{a}{{2R}}$ , $\dfrac{1}{{\sin (A/2)}} = \csc (A/2)$ we get,
$AD = \dfrac{{abc}}{{2R(b + c)}}\csc (A/2)$
As $R = \dfrac{{abc}}{{4\Delta }}$
$AD = \dfrac{{abc.4\Delta }}{{2abc(b + c)}}\csc (A/2)$
Simplifying,
$AD = \dfrac{{2\Delta }}{{b + c}}\csc (A/2)$
So, the correct answer is “Option B” and “Option C”.
Note: Students are likely to make mistakes while substituting the sides using the sine formula, always keep in mind the angle opposite to the side comes in denominator along with the side in the numerator. Making the rearrangements of terms according to our motive is to be done properly for a quick solution. Multiplying and dividing by needed trigonometric terms can cut short the solution into simpler form lines.
* Bisector of an angle A is defined to be the line cutting off A into equal halves which meets the
opposite side. In the below figure, AD is the bisector of angle A. Thus it divides angle A into 2 equal
halves and meets the opposite side BC.
![seo images](https://www.vedantu.com/question-sets/9f8592a4-5e09-4086-b7ac-a77fb4ba5f345196138459157805434.png)
* Bisector of an angle of a triangle divides the opposite side in the ratio of the sides containing the angle.
Implies point D divides BC into the ratio of sides containing angle A which are AC and AB. Thus
$\dfrac{{BD}}{{DC}} = \dfrac{{AC}}{{AB}}$
* Sine rule: Ratios of a side to corresponding opposite angles are equal. $\dfrac{{AB}}{{\sin C}} = \dfrac{{AC}}{{\sin B}} = \dfrac{{BC}}{{\sin A}}$
Complete step-by-step answer:
We start with bisector AD and develop its relation with the interior angles.
Sine rules on the triangle ABD gives the result: $\dfrac{{AD}}{{\sin B}} = \dfrac{{BD}}{{\sin (A/2)}}$
From the property of bisector and sine rule, $BD = \dfrac{c}{{b + c}}a$
$AD = \dfrac{{ca}}{{b + c}}\dfrac{{\sin B}}{{\sin (A/2)}}$
Multiplying numerator and denominator by $2\cos (A/2)$, for simplifying the equation to shorter terms.
$AD = \dfrac{{ca.\sin B.2\cos (A/2)}}{{(b + c)\sin (A/2).2\cos (A/2)}}$
Using the formula $\sin (2\theta ) = 2\sin \theta \cos \theta $ ,
$AD = 2.\dfrac{{ca.\sin B.\cos (A/2)}}{{(b + c)\sin A}}$
Using sine rule: $\dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}}$ we get,
$
AD = 2.\dfrac{{cb\sin B\cos (A/2)}}{{(b + c)\sin B}} \\
= \dfrac{{2cb\cos (A/2)}}{{b + c}} \\
$
From the property of bisector and sine rule,
$AD = \dfrac{{2bc}}{{b + c}}\dfrac{{\cos (A/2)\sin (A/2)}}{{\sin (A/2)}}$
Simplifying,
$AD = \dfrac{{bc}}{{\sin (A/2)}}\dfrac{{\sin A}}{{\sin (A/2)}}$
Using $\sin A = \dfrac{a}{{2R}}$ , $\dfrac{1}{{\sin (A/2)}} = \csc (A/2)$ we get,
$AD = \dfrac{{abc}}{{2R(b + c)}}\csc (A/2)$
As $R = \dfrac{{abc}}{{4\Delta }}$
$AD = \dfrac{{abc.4\Delta }}{{2abc(b + c)}}\csc (A/2)$
Simplifying,
$AD = \dfrac{{2\Delta }}{{b + c}}\csc (A/2)$
So, the correct answer is “Option B” and “Option C”.
Note: Students are likely to make mistakes while substituting the sides using the sine formula, always keep in mind the angle opposite to the side comes in denominator along with the side in the numerator. Making the rearrangements of terms according to our motive is to be done properly for a quick solution. Multiplying and dividing by needed trigonometric terms can cut short the solution into simpler form lines.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)