A triangle has sides with lengths of 3, 9, and 8. What is the radius of the triangles inscribed circle?

Question

A triangle has sides with lengths of 3, 9, and 8.  What is the radius of the triangles inscribed circle?

Jayden Jackson · Accepted Answer

Heron's formula states that
Area = #sqrt(s(s-a)(s-b)(s-c))# Where s is the sum of all sides / 2 (Also known as the semiperimeter)
a,b,c are the side lengths From here, you can just plug in all the values. 
Find that s = 10
Area = #sqrt(10(7)(1)(2))#
Get #sqrt140# Now, we use the formula that says that  Area = semiperimeter * inradius So we have #sqrt140 = 10 * # radius Therefore, the radius is #sqrt140 / 10# We can simplify the radical and cancel out to get #sqrt35 / 5#

James Atkins · Answer

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

$$ r = \frac{2 \times \text{Area}}{ \text{Perimeter}} $$

where $ r $ is the radius of the inscribed circle, Area is the area of the triangle, and Perimeter is the sum of the lengths of the sides of the triangle.

First, we need to find the semi-perimeter of the triangle, which is half of the perimeter:

$$ s = \frac{3 + 9 + 8}{2} = 10 $$

Now, we can use Heron's formula to find the area of the triangle:

$$ \text{Area} = \sqrt{s(s - 3)(s - 9)(s - 8)} $$

$$ \text{Area} = \sqrt{10(10 - 3)(10 - 9)(10 - 8)} $$

$$ \text{Area} = \sqrt{10(7)(1)(2)} $$

$$ \text{Area} = \sqrt{140} $$

$$ \text{Area} ≈ 11.83 $$

Now, we can calculate the radius of the inscribed circle:

$$ r = \frac{2 \times 11.83}{3 + 9 + 8} $$

$$ r = \frac{2 \times 11.83}{20} $$

$$ r ≈ 1.18 $$