Hello math bugs(🐞) and hivers(🐝)
I hope you are strong and stout & doing good in life.

Tiady , I have come up with another interesting triangle (∆ABC) problem. There is a right angle triangle. The right angle(90°) is at point B.The bisectors of other two angles meet at a point **I** inside the triangle. If the perpendiluar on AC from point **I** is 4 cm, you need to find the lenght of IB. Check it the follwoing figure.
![](https://images.ecency.com/DQmZPboMCLdyxioXw2EnCvQGeGqfoKdah1vxYakZwWh4aKP/addtext_04_05_07.35.30.png)

Give it a try first. Then check my solution. 

First, let me point out what concept(s)/postulate(s)/ axiom(s) we need to solve it. In my previous post, I have dealth with all of them except a new concept. Let me put them together.

✅ Angle bisectors of a triangle always meet a point inside the trianglel. The point is called in-centre and distance from that point to the sides is called in-radius. Of course the lenghts are same as they are radius of same cricle. Check it in the following figure.

![](https://images.ecency.com/DQmZrJTYKAzbfTs5vKgSptpXBsCZLdJCjLWDQWWesz98gkm/addtext_04_06_02.50.41.png)





Check details of [In-centre](https://ecency.com/hive-163521/@meta007/knowing-in-circle)

✅✅ The most important point is when two line segments are perpendicular to each other, if another two perpendicular line segments make a quadrilateral with them, it always be an square. Check below what I meant.

![](https://images.ecency.com/DQmRy2RPzwAuirw7WTwY5gjr12VJWPLdPgSpYSrznVzxoMt/addtext_04_06_03.11.52.png)

**Note**: From in-centre two radius drawn to the perpendicular sides of a right angle triangle , always make an square (⬛). No matter what kind of right triangle it is.

✅✅✅ The 3rd point is an easy one. We need to find out the lenght of a diagonal the the square. Here in the problem it will **IB**. Check it below

![](https://images.ecency.com/DQmYa6AhyfVX1hp9mW5cMk3XzrRToGtwh4r1Cc5UPDyh7Px/addtext_04_06_03.29.01.jpg)

If you got these three points, problem will be as easy a eating a sweet  cake. (🍰)

**SOLUTION**:

Now nothing left to solve. Still I'll solve it for you😄. We already know how to find a diaginal. It is **a√2** unit where **a** is the side of the square. Check it 👇

![](https://images.ecency.com/DQmeguNJZCXqAAvVwWrrQxSSjKE4JmebhY6DXn4cuYNjKnn/addtext_04_06_03.47.10.jpg)

Hecne , the required lentgh be 
**IB** = r√2 unit
So, **IB** =4√2 cm. 

**All the pics/figures used are made by me.**  The drawing I made may not be accurate in messurement. So, I request you  to consider the info given only. There may be some silly mistakes while typing or making those figures. Please ingore them if there is any.

I hope you liked my presentention of Today's problem.

Thank you so much for visiting 

Have a nice day

All is well

Regards: @meta007