A triangle has sides with lengths of 8, 4, and 3. What is the radius of the triangles inscribed circle?

Question

A triangle has sides with lengths of 8, 4, and 3.  What is the radius of the triangles inscribed circle?

Evelyn Arboleda · Accepted Answer

The triangle does not exist

For any plane triangle with sides a, b, c The following conditions must be satisfied #a+b>c" "#first condition #a+c>b" "#second condition #b+c>a" "#third condition If we let #a=8# and #b=4# and #c=3# The third condition is FALSE ! Therefore , the triangle does not exist ... God bless...I hope the explanation is useful.

Sadie Allen · Answer

undefined To find the radius of the inscribed circle in a triangle, you can use the formula:

$$ r = \frac{A}{s} $$

Where $ r $ is the radius of the inscribed circle, $ A $ is the area of the triangle, and $ s $ is the semi-perimeter of the triangle. The semi-perimeter $ s $ is calculated as half of the sum of the lengths of the sides of the triangle.

First, calculate the semi-perimeter:

$$ s = \frac{8 + 4 + 3}{2} = \frac{15}{2} = 7.5 $$

Next, use Heron's formula to find the area of the triangle. Heron's formula states that for a triangle with sides of lengths $ a $, $ b $, and $ c $, the area $ A $ is given by:

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

Substitute the given side lengths:

$$ A = \sqrt{7.5(7.5 - 8)(7.5 - 4)(7.5 - 3)} $$

$$ A = \sqrt{7.5 \times (-0.5) \times 3.5 \times 4.5} $$

$$ A = \sqrt{7.5 \times 3.5 \times 4.5} $$

$$ A = \sqrt{94.5} $$

Now, plug the area and semi-perimeter into the formula for the radius of the inscribed circle:

$$ r = \frac{\sqrt{94.5}}{7.5} $$

$$ r \approx \frac{9.72}{7.5} $$

$$ r \approx 1.296 $$

So, the radius of the triangle's inscribed circle is approximately $ 1.296 $ units.