A triangle has sides with lengths of 8, 4, and 6. What is the radius of the triangle's inscribed circle?

Question

2016 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-02-13T03:11:57

We can use semiperimeter and Heron's formula

First, we find the semiperimeter, which is the sum of the sides divided by two.

Therefore, $\frac{8 + 4 + 6}{2} = 9$ $(8+4+6)/2 = 9$ which is the semiperimeter

Now, we can use Heron's formula to find the area, which is
$\sqrt{(s) (s - a) (s - b) (s - c)} = A$ $sqrt((s)(s-a)(s-b)(s-c)) = A$

We just plug in values now
$\sqrt{(9) (9 - 8) (9 - 4) (9 - 6)}$ $sqrt((9)(9-8)(9-4)(9-6))$

which is
$\sqrt{(9) (1) (5) (3)}$ $sqrt((9)(1)(5)(3))$
which is
$\sqrt{135}$ $sqrt135$

Therefore, $\sqrt{135} = A$ $sqrt135 = A$

Now, we can use the formula relating inradii, Area, and semiperimeter

$A = r \cdot s$ $A = r * s$

Since we found two values, we can just plug in to find the third.
$\sqrt{135} = r \cdot 9$ $sqrt135 = r * 9$

$\frac{\sqrt{135}}{9} = r$ $sqrt(135) / 9 = r$

And we are done.