A triangle has corners at $(9 , 2 )$ , $(4 ,7 )$ , and $(5 ,8 )$ . What is the radius of the triangle's inscribed circle?

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A triangle has corners at $(9 , 2 )$ , $(4 ,7 )$ , and $(5 ,8 )$ . What is the radius of the triangle's inscribed circle?

1 Answer

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Answer 1 · 2016-11-28T17:41:11

If $a,b$ and $c$ are the lengths of the sides and $S$ is the area of the triangle the radius of the inscribed circle is given by the formula:

$r=(2S)/(a+b+c)$

Explanation:

The area $S$ can be computed with Eron's formula:

$S=sqrt(p(p-a)(p-b)*(p-c))$

where $p=(a+b+c)/2$ .

Substituting in the formula:

$r=(sqrt(p(p-a)(p-b)*(p-c)))/(p)$

Start calculating the lengths of the sides:

$a=sqrt((x_2-x_1)^2+(y_2-y_1)^2)$
$b=sqrt((x_3-x_2)^2+(y_3-y_2)^2)$
$c=sqrt((x_1-x_3)^2+(y_1-y_3)^2)$

$a = sqrt((9-4)^2+(2-7)^2) = sqrt(25+25) = 5sqrt(2)$
$b= sqrt((5-4)^2+(8-7)^2) =sqrt(1+1) =sqrt(2)$
$c= sqrt((9-5)^2+(8-2)^2) =sqrt(16+36) =2sqrt(13)$

$p=(5sqrt(2)+sqrt(2)+2sqrt(13))/2 = 3sqrt(2)+sqrt(13)$

$r=sqrt((3sqrt(2)+sqrt(13))(3sqrt(2)+sqrt(13)-5sqrt(2))(3sqrt(2)+sqrt(13)-sqrt(2))(3sqrt(2)+sqrt(13)-2sqrt(13)))/(3sqrt(2)+sqrt(13))=sqrt((3sqrt(2)+sqrt(13))(sqrt(13)-2sqrt(2))(2sqrt(2)+sqrt(13))(3sqrt(2)-sqrt(13)))/(3sqrt(2)+sqrt(13))$

and I leave the rest of the calculation :D

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A triangle has corners at $(9 , 2 )$ , $(4 ,7 )$ , and $(5 ,8 )$ . What is the radius of the triangle's inscribed circle?

1 Answer

Explanation:

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Impact of this question

Science

Math

Humanities

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A triangle has corners at (9 , 2 )(9 , 2 ), (4 ,7 )(4 ,7 ), and (5 ,8 )(5 ,8 ). What is the radius of the triangle's inscribed circle?

1 Answer

Explanation:

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A triangle has corners at $(9 , 2 )$ , $(4 ,7 )$ , and $(5 ,8 )$ . What is the radius of the triangle's inscribed circle?