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

Question

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

Sophia Alfred · Accepted Answer

#color(orange)("Area of Circum circle " A_R = pi R^2 = pi * 2.1024^2 = 13.8861 " sq units"#

#"Area of Triangle " = A_T = (a b c) / (4 R)#

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

#a = sqrt((2-3)^2 + (5-1)^2) = sqrt 17 = 4.1231#

#b = sqrt((3-4)^2 + (1-2)^2) = sqrt 2 = 1.4142#

#c = sqrt((4-2)^2 + (2-5)^2) = sqrt 13 = 3.6056#

#"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(4.5715 (4.5715 - 4.1231) (4.5715 - 1.4142) (4.5715 - 3.6056)) = 2.5#

#R = (a b c ) / (4 * A_T) = (sqrt 17 * sqrt 2* sqrt 13) / (4 * 2.5) = 2.1024#

#color(orange)("Area of Circum circle " A_R = pi R^2 = pi * 2.1024^2 = 13.8861 " sq units"#

Caroline Aucoin · Answer

undefined To find the area of the circumscribed circle of a triangle, you can use the formula $A = \frac{abc}{4R}$, where $a$, $b$, and $c$ are the lengths of the triangle's sides and $R$ is the radius of the circumscribed circle.

First, you need to find the lengths of the sides of the triangle using the distance formula between each pair of points.

Then, you can use Heron's formula to find the area of the triangle, which is given by $A = \sqrt{s(s - a)(s - b)(s - c)}$, where $s$ is the semi-perimeter of the triangle.

Once you have the area of the triangle, you can find the radius of the circumscribed circle using $R = \frac{abc}{4A}$, where $a$, $b$, and $c$ are the lengths of the sides of the triangle.

Finally, the area of the circumscribed circle is given by the formula for the area of a circle: $A_{\text{circle}} = \pi R^2$.