Question

what is the value of bisector of right angle ​

Answer 1

Answer:

. A line that splits an angle into two equal angles. ("Bisect" means to divide into two equal parts.)

Answer 2

Step-by-step explanation:

How can we find the length of the angle bisector of a triangle when the sides are given?

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Refer the below image -

Since you are given the three sides of the triangle, a , b and c , first you can find the angle 2C which is bisected. You do this using cosine rule -

c2=a2+b2−2abCos(2C)

Cos(2C)=a2+b2−c22bc

Using half angle formula -

Cos(C)=√[1−Cos(2C)2] …… eqn 1

Again using cosine rule, referring to figure we can write -

n2m2=a2+x2−2axCos(C)b2+x2−2bxCos(C)

According to angle bisector rule -

bm=an

Therefore

a2b2=a2+x2−2axCos(C)b2+x2−2bxCos(C)

a2(b2+x2−2bxCos(C))=b2(a2+x2−2axCos(C)

a2b2+a2x2−2a2bxCos(C)=a2b2+b2x2−2ab2xCos(C)

(a2−b2)x2−2ab(a−b)Cos(C)x=0

x[(a2−b2)x−2ab(a−b)Cos(C)]=0

(a2−b2)x−2ab(a−b)Cos(C)=0 … since x cannot be zero

x=2ab(a−b)Cos(C)a2−b2

x=2ab(a−b)Cos(C)(a+b)(a−b)

x=2abCos(C)a+b

You can use this formula to find the length x of the angle bisector using the value of Cos(C) from eqn 1 above

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There are several methods to find the length of the bisector of an angle of a triangle. I would like to give below one which uses an old theorem of Euclidean geometry known as Stewart- theorem ( named in honour of a Scottish mathematician M. Stewart).

Let ABC be a triangle with side a ,b & c resprespectively opposite to the angle A ,B & C.

That is side BC = a , CA = b and AB = c . Let AD be the angle A bisector meeting the opposite side BC at the point D and dividing the side in the ratio of m : n ( m towards the angle B ) . Then according to Stewart theorem, we must have ;

c^2 ·n + b^2 ·m = a ( m·n +AD^2 ) … … …(1) . Now let us calculate the value of m & n for this expression in terms of the sides lengths. Observe that (make a fig.of the triangle) ;

BD /DC = AB/AC = c/b or BD/(BD+DC) = c/(b+c) or BD/a = c/(b+c) or BD = ac/(b+c) = m. Similarly, DC = ab/(b+c) = n.

Putting the values of m , n. in eq. (1) and solving we find,

AD^2 = d^2 = bc [ 1 - (a/(b+c))^2 ] or

[math]AD = d = sqrt [ (bc) { 1 - ([/math] \dfrac{ a}{(b+c)} )^2 } ] .

Similarly other angle bisector can be found.