A triangle has corners at $(5, 2)$ $(5 , 2 )$ , $(1, 3)$ $(1 ,3 )$ , and $(7, 4)$ $(7 ,4 )$ . What is the radius of the triangle's inscribed circle?

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A triangle has corners at $(5, 2)$ $(5 , 2 )$ , $(1, 3)$ $(1 ,3 )$ , and $(7, 4)$ $(7 ,4 )$ . What is the radius of the triangle's inscribed circle?

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Answer 1 · 2018-06-21T05:13:34

$Radius of incircle r = \frac{A_{t}}{s} = \frac{5.0224}{6.52} = 0.77 units$ $color(blue)("Radius of incircle " r = A_t / s = 5.0224 / 6.52 = 0.77 " units"$

Explanation:

![http://mathibayon.blogspot.com/2015/01/http://derivation-of-formula-for-radius-of-incircle.html](https://useruploads.socratic.org/qByQYJn5SEeAPjtKhB4j_incircle%20radius.png)

$Incircle radius r = \frac{A_{t}}{s}$ $"Incircle radius " r = A_t / s$

$A (5, 2), B (1, 3), C (7, 4)$ $A(5,2), B(1,3), C(7,4)$

$a = \sqrt{{(7 - 1)}^{2} + {(4 - 3)}^{2}} = \sqrt{37}$ $a = sqrt((7-1)^2 + (4-3)^2) = sqrt37$

$b = \sqrt{{(7 - 5)}^{2} + {(4 - 2)}^{2}} = \sqrt{8}$ $b = sqrt((7-5)^2 + (4-2)^2) = sqrt8$

$c = \sqrt{{(5 - 1)}^{2} + {(2 - 3)}^{2}} = \sqrt{17}$ $c = sqrt((5-1)^2 + ( 2-3)^2) = sqrt17$

$Semi-perimeter s = \frac{a + b + c}{2} = \frac{\sqrt{37} + \sqrt{8} + \sqrt{17}}{2} = 6.52$ $"Semi-perimeter " s = (a + b + c) / 2 = (sqrt37 + sqrt8 + sqrt17) / 2 = 6.52$

$A_{t} = \sqrt{s (s - a) s - b} (s - c))$ $"A_t = sqrt(s (s-a) s-b) (s-c))$

$A_{t} = \sqrt{6.52 (6.52 - \sqrt{37}) (6.52 - \sqrt{8}) (6.52 - \sqrt{17})} = 5.0224$ $A_t = sqrt(6.52 (6.52-sqrt37) (6.52 - sqrt8) (6.52 - sqrt17)) =5.0224$

$Radius of incircle r = \frac{A_{t}}{s} = \frac{5.0224}{6.52} = 0.77 units$ $color(blue)("Radius of incircle " r = A_t / s = 5.0224 / 6.52 = 0.77 " units"$

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A triangle has corners at $(5, 2)$ $(5 , 2 )$ , $(1, 3)$ $(1 ,3 )$ , and $(7, 4)$ $(7 ,4 )$ . What is the radius of the triangle's inscribed circle?

1 Answer

Explanation:

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A triangle has corners at (5,2)(5 , 2 ), (1,3)(1 ,3 ), and (7,4)(7 ,4 ). What is the radius of the triangle's inscribed circle?

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A triangle has corners at $(5, 2)$ $(5 , 2 )$ , $(1, 3)$ $(1 ,3 )$ , and $(7, 4)$ $(7 ,4 )$ . What is the radius of the triangle's inscribed circle?