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

Question

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

1 Answer

Impact of this question

2353 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-03-01T06:33:56

Area of the circumcircle

$A_{c} = π R^{2} = π \cdot {12.7032}^{2} = 506.9629$ $color(green)(A_c = pi R^2 = pi * 12.7032^2 = 506.9629$ sq units

Explanation:

$A (5, 7), B (2, 1), C (3, 4)$ $A (5,7), B (2,1), C(3,4)$ . To find the radius of the Incircle.

Using distance formula

$a = \sqrt{{(2 - 3)}^{2} + {(1 - 4)}^{2}} \approx 3.16$ $a = sqrt((2-3)^2 + (1-4)^2) ~~ 3.16$

$b = \sqrt{{(5 - 3)}^{2} + {(7 - 4)}^{2}} \approx 3.6$ $b = sqrt((5-3)^2 + (7-4)^2) ~~ 3.6$

$c = \sqrt{{(5 - 2)}^{2} + {(7 - 1)}^{2}} = 6.7$ $c = sqrt ((5-2)^2 + (7-1)^2) = 6.7$

Semi perimeter of the triangle ABC

$s = \frac{a + b + c}{2} = \frac{3.16 + 3.6 + 6.7}{2} = 6.73$ $s = (a + b + c) / 2 = (3.16 + 3.6 + 6.7) / 2 = 6.73$

enter image source here

Formula for Area of triangle, knowing three sides is

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

$\Rightarrow \sqrt{(6.73 \cdot (6.73 - 3.16) \cdot (6.73 - 3.6) \cdot (6.73 - 6.7)) \approx 1.5}$ $=> sqrt((6.73 * (6.73 - 3.16) * (6.73 - 3.6) * (6.73 - 6.7)) ~~ 1.5$

enter image source here

Radius of the circum circle

$R = \frac{a b c}{4 A_{t}} = \frac{3.16 \cdot 3.6 \cdot 6.7}{4 \cdot 1.5} = 12.7032$ $R = (abc) / (4 A_t) = (3.16 * 3.6 * 6.7) / (4 * 1.5) = color(green)(12.7032$

Area of circumcircle

$A_{c} = π R^{2} = π \cdot {(12.7032)}^{2} \approx 506.9629$ $A_c = pi R^2 = pi * (12.7032)^2 ~~ 506.9629$

Science

Math

Humanities

... and beyond

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

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

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

1 Answer

Explanation:

Related questions

Impact of this question

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