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

Question

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

Claire Akin · Accepted Answer

The area of the triangle's circumscribed circle is#=Delta=41.0096#sq.units

Let , #triangleABC" be the triangle with corners at "#

#A(9,3) , B(4,1) and C(2,4).#

Using Distance formula ,we get

#a=BC=sqrt((4-2)^2+(1-4)^2)=sqrt(4+9)=sqrt13#

#b=CA=sqrt((9-2)^2+(3-4)^2)=sqrt(49+1)=sqrt50#

#c=AB=sqrt((9-4)^2+(3-1)^2)=sqrt(25+4)=sqrt29#

Using cosine Formula ,we get

#cosA=(b^2+c^2-a^2)/(2bc)=(50+29-13)/(2sqrt50sqrt29)=33/(sqrt1450#

We know that,

#sin^2A=1-cos^2A#

#=>sin^2A=1-1089/1450=361/1450#

#=>sinA=19/sqrt1450to[because Ain(0 ^circ,180^circ)]#

Using sine formula:we get

#a/sinA=2R=>R=a/(2sinA)#

#=>R=sqrt13/(2 (19/sqrt1450))=(sqrt13xxsqrt1450)/(2*19)~~3.6130#

So , the area of the triangle's circumscribed circle is:

#Delta=piR^2=pi*(3.6130)^2~~41.0096 ,sq.units#
....................................................................................................

Note:

If we take, #R=3.61# then #A=40.94#

Sophia Alfred · Answer

undefined To find the area of the circumscribed circle of a triangle, we need to find the radius of the circumscribed circle first. The radius $ R $ can be calculated using the formula:

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

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

Then, once you have the radius $ R $, you can find the area of the circumscribed circle using the formula:

$$ Area = \pi R^2 $$

For the given triangle with vertices at $ (9,3), (4,1), $ and $ (2,4) $:

1. Find the side lengths of the triangle using the distance formula between the vertices.
2. Use Heron's formula to find the area of the triangle.
3. Calculate the radius using the formula above.
4. Finally, find the area of the circumscribed circle using the formula above.