A triangle has sides with lengths of 4, 9, and 8. What is the radius of the triangles inscribed circle?

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Answer 1 · 2016-02-09T20:54:00

#1.524#

Refer to the figure below

I created this figure using MS Excel

As the sides of the triangle are 4, 8 and 9:
#x+y=9#
#x+z=8#
#y+z=4# => #z=4-y#
#-> x+4-y=8# => #x-y=4#

Adding the first and last equations
#2x=13# => #x=6.5#

Using the Law of Cosines (#triangle_(ABC)#):
#4^2=9^2+8^2-2*9*8*cos alpha#

#cos alpha=(81+64-16)/144=129/144#

#alpha=26.384^@#

In the right triangle with #x# as cathetus, we can see that
#tan (alpha/2)=r/x#

#r=6.5*tan (26.384^@/2)# => #r=1.524#