A triangle has corners at $(9 ,8 )$ , $(2 ,3 )$ , and $(1 ,4 )$ . What is the area of the triangle's circumscribed circle?

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A triangle has corners at $(9 ,8 )$ , $(2 ,3 )$ , and $(1 ,4 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2017-06-06T10:03:01

The area is $=64.6$

Explanation:

To calculate the area of the circle, we must calculate the radius $r$ of the circle

Let the center of the circle be $O=(a,b)$

Then,

$(9-a)^2+(8-b)^2=r^2$ ....... $(1)$

$(2-a)^2+(3-b)^2=r^2$ .......... $(2)$

$(1-a)^2+(4-b)^2=r^2$ ......... $(3)$

We have $3$ equations with $3$ unknowns

From $(1)$ and $(2)$ , we get

$81-18a+a^2+64-16b+b^2=4-4a+a^2+9-6b+b^2$

$14a+10b=132$

$7a+5b=66$ ............. $(4)$

From $(2)$ and $(3)$ , we get

$4-4a+a^2+9-6b+b^2=1-2a+a^2+16-8b+b^2$

$2a-2b=-4$

$a-b=-2$ .............. $(5)$

From equations $(4)$ and $(5)$ , we get

$7(b-2)+5b=66$

$12b=80$

$b=80/12=20/3$

$a=b-2=20/3-2=14/3$

The center of the circle is $=(14/3,20/3)$

$r^2=(1-a)^2+(4-b)^2=(1-14/3)^2+(4-20/3)^2$

$=11^2/3^2+8^2/3^2$

$=20.56$

The area of the circle is

$A=pi*r^2=20.56*pi=64.6$

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A triangle has corners at $(9 ,8 )$ , $(2 ,3 )$ , and $(1 ,4 )$ . What is the area of the triangle's circumscribed circle?

1 Answer

Explanation:

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Impact of this question

Science

Math

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A triangle has corners at (9 ,8 )(9 ,8 ), (2 ,3 )(2 ,3 ), and (1 ,4 )(1 ,4 ). What is the area of the triangle's circumscribed circle?

1 Answer

Explanation:

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A triangle has corners at $(9 ,8 )$ , $(2 ,3 )$ , and $(1 ,4 )$ . What is the area of the triangle's circumscribed circle?