A triangle has corners at #(9 ,8 )#, #(2 ,3 )#, and #(1 ,4 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(9  ,8   )#, #(2   ,3   )#, and #(1   ,4  )#.  What is the area of the triangle's circumscribed circle?

Maverick Smith · Accepted Answer

The area is #=64.6#

To calculate the area of the circle, we must calculate the radius #r# of the circle

Let the center of the circle be #O=(a,b)#

Then,

#(9-a)^2+(8-b)^2=r^2#.......#(1)#

#(2-a)^2+(3-b)^2=r^2#..........#(2)#

#(1-a)^2+(4-b)^2=r^2#.........#(3)#

We have #3# equations with #3# unknowns

From #(1)# and #(2)#, we get

#81-18a+a^2+64-16b+b^2=4-4a+a^2+9-6b+b^2#

#14a+10b=132#

#7a+5b=66#.............#(4)#

From #(2)# and #(3)#, we get

#4-4a+a^2+9-6b+b^2=1-2a+a^2+16-8b+b^2#

#2a-2b=-4#

#a-b=-2#..............#(5)#

From equations #(4)# and #(5)#, we get

#7(b-2)+5b=66#

#12b=80#

#b=80/12=20/3#

#a=b-2=20/3-2=14/3#

The center of the circle is #=(14/3,20/3)#

#r^2=(1-a)^2+(4-b)^2=(1-14/3)^2+(4-20/3)^2#

#=11^2/3^2+8^2/3^2#

#=20.56#

The area of the circle is

#A=pi*r^2=20.56*pi=64.6#

Aurora Ayala · Answer

undefined To find the area of the circumscribed circle of a triangle, you first need to calculate the circumradius (the radius of the circle that passes through all three vertices of the triangle).

Then, you can use the formula for the area of a circle, which is $ \pi r^2 $, where $ r $ is the radius.

Here's how you can find the circumradius:

1. Calculate the lengths of the sides of the triangle using the distance formula between the given points.
2. Apply Heron's formula to find the area of the triangle using the side lengths.
3. Use the formula for the circumradius of a triangle, which is $ R = \frac{abc}{4A} $, where $ a $, $ b $, and $ c $ are the side lengths of the triangle and $ A $ is its area.

Once you have the circumradius, you can square it and multiply by $ \pi $ to find the area of the circumscribed circle.