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The triangle  \(\Delta ABC\) is an isosceles triangle where \(AB=4\sqrt2\) and \(\angle B\) is a right angle. If  \(L\) is the incenter of \(\Delta ABC\), then what is \(BL\)?

CoolStuffYT  Nov 21, 2018
 #1
avatar+92429 
+1

 

 

We can use the Law  of Sines to solve this

 

Since B is the right angle   and the triangle is isosceles....then  angles  A and C = 45°

 

The incenter of the triangle is the point where the three angle bisectors meet

 

One of the triangles formed by their intersection is ΔBAL

 

Since B is bisected....then angle ABL    =  45°

 

And since A is bisected....angle BAL   = 22.5°

 

And angle ALB   =  180 - 45 - 22.5   = 112.5°

 

So

 

BL                       AB

_______   =      _______

sin BAL            sin ALB

 

BL                    4sqrt(2)

____    =        _________

sin 22.5           sin 112.5

 

 

                     4sqrt (2) sin 22.5

BL =            _______________     ≈    2.34

                          sin 112.5

 

 

 

cool cool cool  

CPhill  Nov 21, 2018

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