A triangle has corners at #(3 ,8 )#, #(5 ,9 )#, and #(8 ,2 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(3  ,8   )#, #(5   ,9   )#, and #(8   ,2  )#.  What is the area of the triangle's circumscribed circle?

Claire Akin · Accepted Answer

Area of the circum-circle #=color(green)48.05#

ABC is the triangl and D,E,F are the midpoints of AB,BC & CA respectively.

Point D has coordinates as #((3+5)/2,(8+9)/2)=(4,17/2)# Similarly, E coordinates #((5+8)/2,(9+2)/2)=(13/2,11/2)# F coordinates #((8+3)/2,(2+8)/2)=(11/2,5)#

Slope of AB #=(9-8)/(5-3)=1/2# Slope of #C’D=-2# where C’ is the circum center and C’D is perpendicular bisector of AB. Circum-center C’ is the meeting point of perpendicular bisectors of the Sides. Equation of C’D is #y-(17/2)=-2*(x-4)# #2y-17=-4x+16# #color(red)(2y+4x=33) #. Eqn (1)

Slope of BC #=9-2/5-8=-(7/3)# Slope ofC’E perpendicular bisector of BC is #=3/7# Equation of C’E is #y-(11/2)=(3/7)(x-(13/2))# #14y-77=6x-39# #color(red)(14y-6x=38)#. Eqn (2)

Solving equations (1) & (2) will give point C’. #6y+12x=99# #28y-12x=76# Adding both equations, #34y=175# #y=175/34# #4x=33-2(175/34)==33-(175/17)=(561-175)/68=386/68=193/34# Coordinates of C’ is #color (red)(193/34,175/34)#

Radius of circum-circle is C’A=C’B=C’C C’A #=sqrt((3-(193/34))^2+(8-(175/34)^2)# #=sqrt(7.16+8.14)=color(blue)3.91#

Verification: C’B #=sqrt(5-(193/34)^2+(9-(175/34))^2)# #=sqrt(0.46+14.85)=color(blue)(3.91)#

Area of circle #=pir^2=(22/7)*(3.91)^2= color(green)(48.05)#

Caroline Aucoin · Answer

undefined To find the area of the triangle's circumscribed circle, you first need to find the circumradius of the triangle. The circumradius (R) of a triangle can be calculated using the formula:

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

Where:
a, b, and c are the lengths of the sides of the triangle, and
A is the area of the triangle.

Once you find the circumradius, you can use the formula for the area of a circle (πR^2) to find the area of the circumscribed circle.

To find the lengths of the sides of the triangle, you can use the distance formula:

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Once you have the lengths of the sides, you can use Heron's formula to find the area of the triangle:

$$ A = \sqrt{s(s - a)(s - b)(s - c)} $$

Where:
s = (a + b + c) / 2
a, b, and c are the lengths of the sides of the triangle.

Finally, plug the values into the formula for the circumradius and then use it to find the area of the circumscribed circle.