A triangle has sides with lengths of 5, 9, and 8. What is the radius of the triangles inscribed circle?

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Answer 1 · 2016-01-21T22:42:29

#1.809#

Refer to the figure below

I created this figure using MS Excel

As the sides of the triangle are 5, 8 and 9:
#x+y=9#
#x+z=8#
#y+z=5# => #z=5-y#
#-> x+5-y=8# => #x-y=3#

Adding the first and last equations
#2x=12# => #x=6#

Using the Law of Cosines:
#5^2=9^2+8^2-2*9*8*cos alpha#

#cos alpha=(81+64-25)/144=120/144=5/6#

#alpha=33.557^@#

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

#r=6*tan (33.557^@/2)# => #r=1.809#

Answer 2 · 2016-01-23T13:27:14

Radius of inscribed circle is #=6/sqrt(11) ~= 1.81#

The radius of a circle inscribed in a triangle is
#color(white)("XXX")r= ("Area"_triangle)/s# where #s# is the semi-perimeter of the triangle.

For a triangle with sides #5, 9, and 8#
#color(white)("XXX")s=11#

Using Heron's formula
#color(white)("XXX")"Area"_triangle = sqrt(s(s-a)(s-b)(s-c))#

#color(white)("XXXXXXX")=sqrt(11(6)(2)(3)) = 6sqrt(11)#

And the required radius is
#color(white)("XXX")(6sqrt(11))/11 = 6/sqrt(11)#