A triangle has sides with lengths of 6, 9, and 8. What is the radius of the triangles inscribed circle?

Question

A triangle has sides with lengths of 6, 9, and 8.  What is the radius of the triangles inscribed circle?

Maverick Smith · Accepted Answer

You should use Heron's formula to find this answer
Find the semiperimeter, which is 11.5
Use Heron's to get Area = #sqrt(11.5*5.5*2.5*3.5)#
inradius is #sqrt(8855)/46# First find the semiperimeter which is the sum of the sides over 2 This is 11.5 Use this to plug into Heron's formula to get #sqrt((23/2)(11/2)(5/2)(7/2))# Simplify to get #sqrt(8855)/4# as the Area Now use the formula Area = inradius * semiperimeter #sqrt(8855)/4# = r * 11.5 #sqrt(8855)# = 46r Thus r = #sqrt(8855)/46#

Theodore Abad · Answer

undefined The radius of the triangle's inscribed circle can be calculated using the formula:

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

Where $ A $ is the area of the triangle, and $ a, b, c $ are the lengths of its sides.

$$ A $$ can be calculated using Heron's formula:

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

Where $ s $ is the semi-perimeter of the triangle, calculated as:

$$ s = \frac{a+b+c}{2} $$

Substituting the given side lengths:

$$ s = \frac{6 + 9 + 8}{2} = 11.5 $$

$$ A = \sqrt{11.5(11.5-6)(11.5-9)(11.5-8)} $$

$$ A = \sqrt{11.5(5.5)(2.5)(3.5)} $$

$$ A = \sqrt{544.8125} $$

$$ A ≈ 23.34 $$

Now, substitute $ A $ and the side lengths into the formula for the radius of the inscribed circle:

$$ r = \frac{2 \times 23.34}{6+9+8} $$

$$ r = \frac{46.68}{23} $$

$$ r ≈ 2.03 $$

Therefore, the radius of the triangle's inscribed circle is approximately 2.03 units.