Angle bisector of triangle; ways to find its value

Hello maths(🐞) and hivers(🐝)
Well come everyone

Today I have come up with an impotrant topic on angle bisector of a triangle. We are going to see the different approaches by which we can obtain the vaules of its length when value of three sides of the triangle are given.

Geometric Approach:

Let's draw the circumcircle of the given triangle ∆ABC. As we are going to find a general formulla, we gonna consider the sides are a, b and c (in unit)which are opposite to ∠A, ∠B and ∠C respectively. Check the following figure.

Now we need to produce the bisector so that it touches the circumference at E and then let's draw CE as you can see it in the following figure.

You must be known to the fact that in case of angle bisector of a triangle that the ratio two sides of the angle which is bisected is equal to the ratio of the line segments of opposite side into which the biseotor devides it. Check the following figure.

We need the values of BD and CD to find AD, the bisector. We see it later.

Now from ∆ABD and ∆AEC WE CAN HAVE
∠ABC = ∠CEA [angles on the same arc]
and ∠BAE and ∠CAE [Given]
So, ∆ABD and ∆AEC are similiar

The triangle with light green colour is similiar to the blue coloured triangle.

Let's see what can be concluded from the avaiable facts. Check the figure below.👇

We got a relation but the issue is we do not know the value of ED. So, let's take another fact into consideration. It can be given by AD × ED = BD × CD. I am not going to prove it here. You can check my previous post here.

Let check the final relation of a bisector to the other sides of a triangle in the figure below 👇

Solution:

See, we need to know the value of AB, AC, BD and CD. Initially, in the problem it was given that AB c= 6 cm , AC = b =7 cm and BC = a = 8 cm.. And we already got what are BD and CD denoted by. So let's go to the solving part now. Check the figure below. 👇

Trogonometric Approach: We can use both sine rule and cosine rule to get the value of the bisector AD. The post is alreay very long. So I am not doing it right now. With given hints, you can try it.

In next post I'll show you both of the approaches and also the stewart law with proof. If we know Stewart's law, we need not to follow any of the approach I mentioned above.

All the figures are made by me using android application. Please ingore the silly mistakes if there is any. Just consider the given data.

That's all for today.

I hope you got this post interesting and useful.

Thank you so much for visiting.

Have a great day

All is well

Regards: @meta007