A triangle has corners at $(1 ,5 )$ , $(7 ,9 )$ , and $(4 ,2 )$ . What is the area of the triangle's circumscribed circle?

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A triangle has corners at $(1 ,5 )$ , $(7 ,9 )$ , and $(4 ,2 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2017-07-27T07:32:46

The area of the circumscribed circle is $=47.38u^2$

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,

$(1-a)^2+(5-b)^2=r^2$ ....... $(1)$

$(7-a)^2+(9-b)^2=r^2$ .......... $(2)$

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

We have $3$ equations with $3$ unknowns

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

$1-2a+a^2+25-10b+b^2=49-14a+a^2+81-18b+b^2$

$12a+8b=130-26=104$

$3a+2b=26$ ............. $(4)$

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

$16-8a+a^2+4-4b+b^2=49-14a+a^2+81-18b+b^2$

$6a+14b=130-20=110$

$3a+7b=55$ .............. $(5)$

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

$26-2b=55-7b$ , $=>$ , $5b=29$ , $b=29/5$

$3a=26-2b=26-58/5=72/5$ , $=>$ , $a=24/5$

The center of the circle is $=(24/5,29/5)$

$r^2=(1-24/5)^2+(5-29/5)^2=(-19/5)^2+(-4/5)^2$

$=361/25+16/25$

$=377/25$

The area of the circle is

$A=pi*r^2=pi*377/25=47.38u^2$

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A triangle has corners at $(1 ,5 )$ , $(7 ,9 )$ , and $(4 ,2 )$ . What is the area of the triangle's circumscribed circle?

1 Answer

Explanation:

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Impact of this question

Science

Math

Humanities

... and beyond

A triangle has corners at (1 ,5 )(1 ,5 ), (7 ,9 )(7 ,9 ), and (4 ,2 )(4 ,2 ). What is the area of the triangle's circumscribed circle?

1 Answer

Explanation:

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A triangle has corners at $(1 ,5 )$ , $(7 ,9 )$ , and $(4 ,2 )$ . What is the area of the triangle's circumscribed circle?