A triangle has corners at #(5 ,2 )#, #(8 ,1 )#, and #(3 ,4 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(5  ,2   )#, #(8   ,1  )#, and #(3   ,4  )#.  What is the area of the triangle's circumscribed circle?

Wyatt Anderson · Accepted Answer

#"area is 42.5"pi#

#"the points A,B,C are on the circle."#
#"we can write the fallowing equations"#

#A(5,2)#

#x^2+y^2+ax+bx+c=0#
#5^2+2^2+5a+2b+c=0#
#25+4+5a+2b+c=0#

#5a+2b+c=-29" (1)"#

#B(8,1)#

#x^2+y^2+ax+bx+c=0#
#8^2+1^2+8a+b+c=0#
#64+1+8a+b+c=0#

#8a+b+c=-65" (2)"#

#C(3,4)#

#x^2+y^2+ax+bx+c=0#
#3^2+4^2+3a+4b+c=0#
#9+16+3a+4b+c=0#

#3a+4b+c=-25" (3)"#

#"let's add the equation (2) to (3)"#
#11a+5b+2c=-90" (4)"#

#"expand the equation (1) by 2"#
#10a+4b+2c=-58" (5)"#

#"subtract the equation (5) from (4)"#

#a+b=-32" (5)"#

#"subtract the equation (1) from (2)"#

#3a-b=-36" (6)"#

#"now add (5) to (6)"#

#4a=-68#

#color(red)(a=-17)#

#"use (5)"#

#-17+b=-32#

#b=-32+17#

#color(red)(b=-15)#

#"use (2)"#

#8(-17)-15+c=-65#

#-136-15+c=-65#

#c=-65+151#

#color(red)(c=86)#

#"r:radius of circle"#

#r=sqrt(a^2+b^2-4c)/2#

#r=sqrt(289+225-344)/2#

#r=sqrt(514-344)#

#r=sqrt(170)/2#

#r^2=170/4=42.5#

#area=pi*r^2#

#area=42.5pi#

Caroline Aucoin · Answer

undefined To find the area of the circumscribed circle of a triangle, you first need to find the circumradius, which is the radius of the circle circumscribed around the triangle.

The circumradius $ R $ of a triangle with side lengths $ a $, $ b $, and $ c $ can be found using the formula:

$$ R = \frac{abc}{4A} $$

Where $ A $ is the area of the triangle.

To find the area $ A $ of the triangle formed by the given points, you can use the Shoelace formula or another method to calculate the area of a triangle given its vertices.

Once you have the area $ A $, you can find the circumradius $ R $ using the formula above.

Finally, once you have the circumradius $ R $, the area of the circumscribed circle is given by:

$$ \text{Area} = \pi R^2 $$

Where $ \pi $ is the mathematical constant pi.

So, you would calculate the area of the circumscribed circle using the formula above with the calculated circumradius $ R $.