A triangle has sides with lengths of 6, 4, and 3. What is the radius of the triangles inscribed circle?

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A triangle has sides with lengths of 6, 4, and 3. What is the radius of the triangles inscribed circle?

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Answer 1 · 2016-06-07T23:55:44

$r = \sqrt{\frac{(p - a) (p - b) (p - c)}{p}}$ $r=sqrt(((p-a)(p-b)(p-c))/p)$ with $p = \frac{a + b + c}{2}$ $p = (a+b+c)/2$
$r = 0.820413$ $r = 0.820413$

Explanation:

From Heron's formula, the area of a triangle giving their sides $a, b, c$ $a,b,c$ is

$A = \sqrt{p (p - a) (p - b) (p - c)}$ $A=sqrt(p(p-a)(p-b)(p-c))$ with $p = \frac{a + b + c}{2}$ $p = (a+b+c)/2$

Let $o$ $o$ be the triangle orthocenter, then the distance between each side and $o$ $o$ is $r$ $r$ which is the inscribed circle radius. So

$A = \frac{a \times r}{2} + \frac{b \times r}{2} + \frac{c \times r}{2}$ $A = (a xx r)/2+(b xx r)/2 + (c xx r)/2$ then
$r (\frac{a + b + c}{2}) = A = r \times p$ $r((a+b+c)/2)=A = r xx p$

Finally

$r = \frac{\sqrt{p (p - a) (p - b) (p - c)}}{p} = \sqrt{\frac{(p - a) (p - b) (p - c)}{p}} = 0.820413$ $r = (sqrt(p(p-a)(p-b)(p-c)))/p=sqrt(((p-a)(p-b)(p-c))/p) = 0.820413$

Answer 2 · 2016-06-13T21:39:14

$≅ 0.820$ $~=0.820$

Explanation:

I created this figure using MS Excel

Suppose
$A B = 6$ $AB=6$
$B C = 4$ $BC=4$
$C A = 2$ $CA=2$

$m + l = 6$ $m+l=6$ [1]
$l + n = 4$ $l+n=4$ [2]
$m + n = 3$ $m+n=3$ [3]

[3]-[2]
$m - l = - 1$ $m-l=-1$ [4]

[4]+[1]
$2 m = 5$ $2m=5$ => $m = \frac{5}{2}$ $m=5/2$
$\to l = \frac{7}{2}$ $-> l=7/2$
$\to n = \frac{1}{2}$ $-> n=1/2$

Applying law of cosines:
$A B^{2} = B C^{2} + C A^{2} - 2 \cdot B C \cdot C A \cdot cos (A ˆ C B)$ $AB^2=BC^2+CA^2-2*BC*CA*cos(A hat C B)$
$36 = 16 + 9 - 2 \cdot 4 \cdot 3 cos (A ˆ C B)$ $36=16+9-2*4*3cos(AhatCB)$
$24 cos (A ˆ C B) = - 11$ $24cos(AhatCB)=-11$ => $A ˆ C B ≅ {117.28}^{\circ}$ $AhatCB~=117.28^@$

$tan (\frac{A ˆ C B}{2}) = \frac{r}{n}$ $tan((AhatCB)/2)=r/n$
$r = \frac{1}{2} \cdot tan (\frac{{117.28}^{\circ}}{2}) = 0.820$ $r=1/2*tan(117.28^@/2)=0.820$

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A triangle has sides with lengths of 6, 4, and 3. What is the radius of the triangles inscribed circle?

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