A triangle has corners at #(6 ,4 )#, #(8 ,2 )#, and #(3 ,6 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(6  ,4   )#, #(8   ,2   )#, and #(3   ,6  )#.  What is the area of the triangle's circumscribed circle?

James Atkins · Accepted Answer

#"Area" = 533/2pi#

The standard Cartesian form for the equation of a circle is:

#(x-h)^2+(y-k)^2=r^2" [1]"#

where #(x,y)# is any point on the circle, #(h,k)# is the center point, and #r# is the radius.

The points #(6,4)#, #(8,2)#, and #(3,6)# must lie on the circumscribed circle, therefore, we can use these points to write 3 unique equations:

#(6-h)^2+(4-k)^2=r^2" [2]"# #(8-h)^2+(2-k)^2=r^2" [3]"# #(3-h)^2+(6-k)^2=r^2" [4]"#

Expand the squares:

#36-12h+h^2+16-8k+k^2=r^2" [2.1]"# #64-16h+h^2+4-4k+k^2=r^2" [3.1]"# #9-6h+h^2+36-12k+k^2=r^2" [4.1]"#

Subtract equation [4.1] from equation [2.1]:

#7-6h+4k=0" [5]"#

Subtract equation [4.1] from equation [3.1]:

#23-10h+8k=0" [6]"#

Multiply equation [5] by -2 and add it to equation [6]:

#9+2h=0#

#h = -9/2#

Substitute #h = -9/2# into equation [5] and then solve for #k#:

#7-6(-9/2)+4k=0#

#34 + 4k=0#

#k = -17/2#

Substitute #h = -9/2# and #k = -17/2# into equation [2] and the solve for #r^2#:

#(6+9/2)^2+(4+17/2)^2=r^2#

#(12/2+9/2)^2+(8/2+17/2)^2=r^2#

#(21/2)^2+(25/2)^2 = r^2#

#r^2 = 533/2#

The area of the circle is:

#"Area" = pir^2#

#"Area" = 533/2pi#

Sadie Adams · Answer

undefined To find the area of the circumscribed circle of a triangle, you can use the formula:

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

Where $ A $ is the area of the triangle, $ 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:

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

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

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

Where $ s $ is the semi-perimeter of the triangle, given by $ s = \frac{a + b + c}{2} $.

After finding the area of the triangle, you can calculate the radius of the circumscribed circle using the formula:

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

Finally, you can use the radius $ R $ to find the area of the circumscribed circle using the formula for the area of a circle:

$$ A_{circle} = \pi R^2 $$

By plugging in the values you've calculated, you can find the area of the circumscribed circle of the triangle.