A triangle has corners at #(2 ,8 )#, #(3 ,9 )#, and #(4 ,7 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(2  ,8   )#, #(3   ,9   )#, and #(4   ,7  )#.  What is the area of the triangle's circumscribed circle?

Avery Allen · Accepted Answer

#A = (50pi)/36#

Using the standard equation of a circle, #(x - h)^2 + (y - k)^2 = r^2# we can use the 3 given points to write 3 equations:

[1] #(2 - h)^2 + (8 - k)^2 = r^2# [2] #(3 - h)^2 + (9 - k)^2 = r^2# [3] #(4 - h)^2 + (7 - k)^2 = r^2#

Because #r^2 = r^2# we can set the left side of equation [1] equal to the left side of equation [2]:

#(2 - h)^2 + (8 - k)^2 = (3 - h)^2 + (9 - k)^2#

And the same for equations [1] and [3]:

#(2 - h)^2 + (8 - k)^2 = (4 - h)^2 + (7 - k)^2#

Expand the squares for both equations, using the pattern #(a - b)^2 = a^2 - 2ab + b^2#:

#4 - 4h + h^2 + 64 - 16k + k^2 = 9 -6h + h^2 + 81 -18k+ k^2# #4 - 4h + h^2 + 64 - 16k + k^2 = 16 -8h + h^2 + 49 -14k + k^2#

The square terms cancel:

#4 - 4h + 64 - 16k = 9 - 6h + 81 -18k# #4 - 4h + 64 - 16k = 16 -8h + 49 -14k#

Collect all of the constant terms on the left:

#-4h -16k = -6h -18k + 22 # #-4h -16k = -8h-14k - 3#

Collect all the h terms on the right:

#-16k = -2h -18k + 22 # #-16k = -4h-14k - 3#

Collect all of the k terms on the left:

[4] #2k = -2h + 22 # [5] #-2k = -4h - 3#

Divide equation 4 by 2 and equation 5 by -2:

[6] #k = -h + 11 # [7] #k = 2h + 3/2#

Because #k = k# set the right side of equation [7] equal to the right side of equation [6]

#2h + 3/2 = -h + 11 #

#3h = 11 - 3/2 #

#h = 19/6#

Substitute #19/6# for h in equation [6]

#k = -19/6 + 11#

#k = 47/6#

To find the value of #r^2#, substitute #19/6# for h and #47/6# for k in equation [1]

#(2 - 19/6)^2 + (8 - 47/6)^2 = r^2#

#(-7/6)^2 + (1/6)^2 = r^2#

#49/36 + 1/36 = r^2#

#r^2 = 50/36#

The area, A, of the circle is #pir^2#

#A = (50pi)/36#

Wyatt Anderson · Answer

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

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

Where $a$, $b$, and $c$ are the lengths of the sides of the triangle, and $R$ is the radius of the circumscribed circle.

First, calculate the lengths of the sides of the triangle using the distance formula:

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

$b = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}$

$c = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2}$

Then, use Heron's formula to find the area of the triangle:

$s = \frac{a + b + c}{2}$

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

Once you have the area of the triangle, you can find the radius of the circumscribed circle using the formula:

$R = \frac{abc}{4Area_{triangle}}$

Finally, plug in the values to find the area of the circumscribed circle using the formula:

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

Calculating these values will give you the area of the triangle's circumscribed circle.