A triangle has corners at #(5 ,6 )#, #(1 ,3 )#, and #(6 ,5 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(5  ,6   )#, #(1   ,3   )#, and #(6   ,5  )#.  What is the area of the triangle's circumscribed circle?

Wyatt Anderson · Accepted Answer

Area of Circumcircle #A_C = pi 2.7199^2 = color(green)(23.2411)# sq units

Slope of AB = (y2-y1) / (x2-x1) = (3-6) / (1-5) = 3/4#

Slope of #OM_C# perpendicular to AB is

#m_(M_C) = -1 / (3/4) = -4/3#

Cordinates of #M_c = (5+1)/2, (6 + 3)/2 = (3, 9/2)#

Equation of #OM_C# is

#y - (9/2) = -(4/3) (x - 3)#

#6y - 27 = -8x + 24#

#6y + 8x = 51# Eqn (1)

Slope of BC = (y2-y1) / (x2-x1) = (5-3) / (6-1) = 2/5#

Slope of #OM_A# perpendicular to BC is

#m_(M_A) = -1 / (2/5) = -5/2#

Cordinates of #M_c = (6+1)/2, (5 + 3)/2 = (7/2, 4)#

Equation of #OM_A# is

#y - 4 = -(5/2) (x - (7/2))#

#y - 4 = -(5/4) (2x - 7)#

#4y - 16 = -10x + 35#

#4y + 10x = 51# Eqn (2)

Solving Eqns (1), (2), we get circumcenter coordinates.

#O(51/14, 51/14)#

Radius of circumcenter R is the distance between the circumecenter O and any one of the vertices.

#R = sqrt((5- (51/14))^2 + (6 - (51/14))^2) = 2.7199#

Area of Circumcircle #A_C = pi 2.7199^2 = 23.2411# sq units

Sophia Alfred · Answer

undefined To find the area of the circumscribed circle of a triangle, we need to determine the radius of the circle first, which is the distance from the circumcenter (center of the circumscribed circle) to any of the triangle's vertices. Then, we can use the formula for the area of a circle ($A = \pi r^2$) to find the area of the circumscribed circle.

Would you like assistance in finding the circumcenter and subsequently calculating the radius and area of the circumscribed circle?