A triangle has corners at $(2, 4)$ $(2 ,4 )$ , $(6, 5)$ $(6 ,5 )$ , and $(3, 3)$ $(3 ,3 )$ . What is the area of the triangle's circumscribed circle?

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A triangle has corners at $(2, 4)$ $(2 ,4 )$ , $(6, 5)$ $(6 ,5 )$ , and $(3, 3)$ $(3 ,3 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2017-10-02T06:18:02

See the answer below...

Explanation:

Suppose the circumcentre of the triangle is $(x, y)$ $(x,y)$

$1st STEP$
Hence the distance of the point of the circumcentre from the corners of the triangle will be same...
So we can write ,
$\sqrt{{(x - 2)}^{2} + {(y - 4)}^{2}} = \sqrt{{(x - 6)}^{2} + {(y - 5)}^{2}} = \sqrt{{(x - 3)}^{2} + {(y - 3)}^{2}}$ $sqrt((x-2)^2+(y-4)^2)=sqrt((x-6)^2+(y-5)^2)=sqrt((x-3)^2+(y-3)^2)$
[Squaring each part and removing $x^{2}$ $x^2$ and $y^{2}$ $y^2$ from each part]

$\Rightarrow - 4 x + 4 - 8 y + 16 = - 12 x + 36 - 10 y + 25 = - 6 x + 9 - 6 y + 9$ $=>-4x+4-8y+16=-12x+36-10y+25=-6x+9-6y+9$
Hence we get equations ...
$i)$ $i)$ $8 x + 2 y = 41$ $8x+2y=41$ [From 1st and 2nd equation ]
$i i) 6 x + 4 y = 43$ $ii) 6x+4y=43$ [From 2nd and 3rd equation]

From these equations we get
$16 x + 4 y - 6 x - 4 y = 82 - 43$ $16x+4y-6x-4y=82-43$
$\Rightarrow x = 3.9$ $=>x=3.9$
Similarly we get the value of $y = \frac{41 - 8 \times 3.9}{2} = 4.9$ $y=(41-8xx3.9)/2=4.9$

Thus we can determine the value of the radius of the circle...
$\sqrt{{(3.9 - 2)}^{2} + {(4.9 - 2)}^{2}} = 2.1 u n i t$ $sqrt((3.9-2)^2+(4.9-2)^2)=2.1 unit$ (approx)
Hence the area of the circle is $π \times r^{2} = 13.86 u n i t^{2}$ $pixxr^2=13.86 unit^2$ (approx)

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A triangle has corners at $(2, 4)$ $(2 ,4 )$ , $(6, 5)$ $(6 ,5 )$ , and $(3, 3)$ $(3 ,3 )$ . What is the area of the triangle's circumscribed circle?

1 Answer

Explanation:

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A triangle has corners at (2 ,4 )(2,4)(2 ,4 ), (6 ,5 )(6,5)(6 ,5 ), and (3 ,3 )(3,3)(3 ,3 ). What is the area of the triangle's circumscribed circle?

1 Answer

Explanation:

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A triangle has corners at $(2, 4)$ $(2 ,4 )$ , $(6, 5)$ $(6 ,5 )$ , and $(3, 3)$ $(3 ,3 )$ . What is the area of the triangle's circumscribed circle?