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

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

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Answer 1 · 2018-07-10T15:51:58

$Area of Circum circle A_{R} = π R^{2} = π \cdot {2.1024}^{2} = 13.8861 sq units$ $color(orange)("Area of Circum circle " A_R = pi R^2 = pi * 2.1024^2 = 13.8861 " sq units"$

Explanation:

![http://mathibayon.blogspot.com/2015/01/ derivation-of-formula-for-radius-of-circumcircle.html

$Area of Triangle = A_{T} = \frac{a b c}{4 R}$ $"Area of Triangle " = A_T = (a b c) / (4 R)$

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

$a = \sqrt{{(2 - 3)}^{2} + {(5 - 1)}^{2}} = \sqrt{17} = 4.1231$ $a = sqrt((2-3)^2 + (5-1)^2) = sqrt 17 = 4.1231$

$b = \sqrt{{(3 - 4)}^{2} + {(1 - 2)}^{2}} = \sqrt{2} = 1.4142$ $b = sqrt((3-4)^2 + (1-2)^2) = sqrt 2 = 1.4142$

$c = \sqrt{{(4 - 2)}^{2} + {(2 - 5)}^{2}} = \sqrt{13} = 3.6056$ $c = sqrt((4-2)^2 + (2-5)^2) = sqrt 13 = 3.6056$

$Semi perimeter of the triangle s = \frac{a + b + c}{2} = 4.5715$ $"Semi perimeter of the triangle " s = (a + b + c ) / 2 = 4.5715$

$A_{T} = \sqrt{s (s - a) (s - b) (s - c)}$ $A_T = sqrt (s (s-a) (s-b) (s - c))$

$A_{T} = \sqrt{4.5715 (4.5715 - 4.1231) (4.5715 - 1.4142) (4.5715 - 3.6056)} = 2.5$ $A_T = sqrt(4.5715 (4.5715 - 4.1231) (4.5715 - 1.4142) (4.5715 - 3.6056)) = 2.5$

$R = \frac{a b c}{4 \cdot A_{T}} = \frac{\sqrt{17} \cdot \sqrt{2} \cdot \sqrt{13}}{4 \cdot 2.5} = 2.1024$ $R = (a b c ) / (4 * A_T) = (sqrt 17 * sqrt 2* sqrt 13) / (4 * 2.5) = 2.1024$

$Area of Circum circle A_{R} = π R^{2} = π \cdot {2.1024}^{2} = 13.8861 sq units$ $color(orange)("Area of Circum circle " A_R = pi R^2 = pi * 2.1024^2 = 13.8861 " sq units"$

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

1 Answer

Explanation:

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

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Explanation:

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