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

Answer 1

Evelyn Arboleda

#1.524#

Refer to the figure below

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#

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Answer 2

Theodore Abad

The radius ( r ) of the triangle's inscribed circle is given by the formula:

[ r = \frac{2A}{a + b + c} ]

where ( A ) is the area of the triangle, ( a ), ( b ), and ( c ) are the lengths of its sides. The area ( A ) can be calculated using Heron's formula:

[ A = \sqrt{s(s - a)(s - b)(s - c)} ]

where ( s ) is the semiperimeter of the triangle given by:

[ s = \frac{a + b + c}{2} ]

Substitute the given side lengths ( a = 4 ), ( b = 9 ), and ( c = 8 ) into the formulas to find ( r ).

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