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

2 Answers
Jan 21, 2016

1.809

Explanation:

Refer to the figure below

I created this figure using MS ExcelI 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

Jan 23, 2016

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

Explanation:

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)