A triangle has corners at $(2, 7)$ $(2 ,7 )$ , $(3, 1)$ $(3 ,1 )$ , and $(8, 9)$ $(8 ,9 )$ . What is the area of the triangle's circumscribed circle?

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A triangle has corners at $(2, 7)$ $(2 ,7 )$ , $(3, 1)$ $(3 ,1 )$ , and $(8, 9)$ $(8 ,9 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2016-10-19T12:01:54

Area of inscribed circle is (approx.) $9.5096$ $9.5096$ (sq.units)

Explanation:

If $A = (2, 7), B = (3, 1), and C = (8, 9)$ $A=(2,7), B=(3,1), and C=(8,9)$
then
$XXX | A B | = \sqrt{{(3 - 2)}^{2} + {(1 - 7)}^{2}} = \sqrt{37} \approx 6.0828$ $color(white)("XXX")abs(AB)=sqrt((3-2)^2+(1-7)^2)=sqrt(37)~~6.0828$
$XXX | B C | = \sqrt{{(8 - 3)}^{2} + {(9 - 1)}^{2}} = \sqrt{89} \approx 9.4340$ $color(white)("XXX")abs(BC)=sqrt((8-3)^2+(9-1)^2)=sqrt(89)~~9.4340$
$XXX | C A | = \sqrt{{(2 - 8)}^{2} + {(7 - 9)}^{2}} = \sqrt{40} \approx 6.3246$ $color(white)("XXX")abs(CA)=sqrt((2-8)^2+(7-9)^2)=sqrt(40)~~6.3246$

The perimeter, $p_{△}$ $p_triangle$ m of the triangle is
$XXX p_{△} = | A B | + | B C | + | C A | \approx 21.8413$ $color(white)("XXX")"p_triangle=abs(AB)+abs(BC)+abs(CA)~~21.8413$

The semi-perimeter< $s_{△}$ $s_triangle$ , of the triangle is
$XXX s_{△} = \frac{p_{△}}{2} \approx 10.9207$ $color(white)("XXX")s_triangle = (p_triangle)/2~~10.9207$

The area of the triangle, ${area}_{△}$ $"area"_triangle$ can be calculated geometrically or using Heron's Formula:
$XXX {area}_{△} = \sqrt{s (s - a) (s - b) (s - c)} \approx 19$ $color(white)("XXX")"area"_triangle = sqrt(s(s-a)(s-b)(s-c))~~19$

The radius, $r_{\circ}$ $r_circ$ of a circle inscribed in a triangle is given by the formula
$XXX r_{\circ} = \frac{{area}_{△}}{s} \approx \frac{19}{10.9207} \approx 1.7398$ $color(white)("XXX")r_circ=("area"_triangle)/s ~~ 19/10.9207 ~~1.7398$

The area, ${area}_{\circ}$ $"area"_circ$ , of a circle inscribed in a triangle is given by the formula
$XXX {area}_{\circ} = π \cdot r_{\circ}^{2} \approx 9.5096$ $color(white)("XXX")"area"_circ = pi * r_circ ^2 ~~9.5096$

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A triangle has corners at $(2, 7)$ $(2 ,7 )$ , $(3, 1)$ $(3 ,1 )$ , and $(8, 9)$ $(8 ,9 )$ . What is the area of the triangle's circumscribed circle?

1 Answer

Explanation:

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A triangle has corners at (2,7)(2 ,7 ), (3,1)(3 ,1 ), and (8,9)(8 ,9 ). What is the area of the triangle's circumscribed circle?

1 Answer

Explanation:

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A triangle has corners at $(2, 7)$ $(2 ,7 )$ , $(3, 1)$ $(3 ,1 )$ , and $(8, 9)$ $(8 ,9 )$ . What is the area of the triangle's circumscribed circle?