A triangle has corners at #(5 ,8 )#, #(2 ,9 )#, and #(7 ,3 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(5  ,8   )#, #(2   ,9   )#, and #(7   ,3  )#.  What is the area of the triangle's circumscribed circle?

Sadie Allen · Accepted Answer

#(x-51/26)²+(y-101/26)²=(sqrt(8845/2)/13)²# The circumference equation with center in #(a,b)# and radius #r# is given by #(x-a)²+(y-b)²=r²#. Given three non aligned points, #P_1,P_2# and #P_3# an unique circumference passes through them. If the circumference pass through those points, the points must verify the circumference equation. #P_1->(x_1-a)²+(y_1-b)²=r²# #P_2->(x_2-a)²+(y_2-b)²=r²# #P_2->(x_3-a)²+(y_3-b)²=r²# We have then three equations in the unknowns #(a,b,c)# They read: #P_1->89 - 10 a + a^2 - 16 b + b^2 =r^2# #P_2->85 - 4 a + a^2 - 18 b + b^2 = r^2# #P_3->58 - 14 a + a^2 - 6 b + b^2 =r^2# We can easily solve those equations taking instead #P_1-P_2# and #P_2-P_3# and solving for #(a,b)# So we obtain the solution: #a=51/26,b=101/26,r=sqrt(8845/2)/13#

Theodore Abad · Answer

undefined To find the area of the circumscribed circle of a triangle, we first need to find the circumradius $ R $ of the circle. Then, we can use the formula $ A = \pi R^2 $, where $ A $ is the area of the circle.

The circumradius $ R $ of a triangle can be calculated using the formula:

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

where $ a $, $ b $, and $ c $ are the side lengths of the triangle, and $ A $ is the area of the triangle.

To find the side lengths of the triangle, we can use the distance formula between the given points:

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

Using this formula, we can find the lengths of the three sides of the triangle:

$$ d_1 = \sqrt{(2 - 5)^2 + (9 - 8)^2} $$
$$ d_2 = \sqrt{(7 - 2)^2 + (3 - 9)^2} $$
$$ d_3 = \sqrt{(7 - 5)^2 + (3 - 8)^2} $$

Then, we calculate the semiperimeter $ s $ of the triangle:

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

where $ a $, $ b $, and $ c $ are the side lengths of the triangle.

Now, we can use Heron's formula to find the area $ A $ of the triangle:

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

Once we have the area of the triangle, we can calculate the circumradius $ R $ using the formula mentioned earlier. Finally, we can use the formula $ A = \pi R^2 $ to find the area of the circumscribed circle.