A triangle has corners at $(5, 2)$ $(5 ,2 )$ , $(8, 1)$ $(8 ,1 )$ , and $(3, 4)$ $(3 ,4 )$ . What is the area of the triangle's circumscribed circle?

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A triangle has corners at $(5, 2)$ $(5 ,2 )$ , $(8, 1)$ $(8 ,1 )$ , and $(3, 4)$ $(3 ,4 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2016-12-07T13:33:27

$area is 42.5 π$ $"area is 42.5"pi$

Explanation:

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$the points A,B,C are on the circle.$ $"the points A,B,C are on the circle."$
$we can write the fallowing equations$ $"we can write the fallowing equations"$

$A (5, 2)$ $A(5,2)$

$x^{2} + y^{2} + a x + b x + c = 0$ $x^2+y^2+ax+bx+c=0$
$5^{2} + 2^{2} + 5 a + 2 b + c = 0$ $5^2+2^2+5a+2b+c=0$
$25 + 4 + 5 a + 2 b + c = 0$ $25+4+5a+2b+c=0$

$5 a + 2 b + c = - 29 (1)$ $5a+2b+c=-29" (1)"$

$B (8, 1)$ $B(8,1)$

$x^{2} + y^{2} + a x + b x + c = 0$ $x^2+y^2+ax+bx+c=0$
$8^{2} + 1^{2} + 8 a + b + c = 0$ $8^2+1^2+8a+b+c=0$
$64 + 1 + 8 a + b + c = 0$ $64+1+8a+b+c=0$

$8 a + b + c = - 65 (2)$ $8a+b+c=-65" (2)"$

$C (3, 4)$ $C(3,4)$

$x^{2} + y^{2} + a x + b x + c = 0$ $x^2+y^2+ax+bx+c=0$
$3^{2} + 4^{2} + 3 a + 4 b + c = 0$ $3^2+4^2+3a+4b+c=0$
$9 + 16 + 3 a + 4 b + c = 0$ $9+16+3a+4b+c=0$

$3 a + 4 b + c = - 25 (3)$ $3a+4b+c=-25" (3)"$

$let's add the equation (2) to (3)$ $"let's add the equation (2) to (3)"$
$11 a + 5 b + 2 c = - 90 (4)$ $11a+5b+2c=-90" (4)"$

$expand the equation (1) by 2$ $"expand the equation (1) by 2"$
$10 a + 4 b + 2 c = - 58 (5)$ $10a+4b+2c=-58" (5)"$

$subtract the equation (5) from (4)$ $"subtract the equation (5) from (4)"$

$a + b = - 32 (5)$ $a+b=-32" (5)"$

$subtract the equation (1) from (2)$ $"subtract the equation (1) from (2)"$

$3 a - b = - 36 (6)$ $3a-b=-36" (6)"$

$now add (5) to (6)$ $"now add (5) to (6)"$

$4 a = - 68$ $4a=-68$

$a = - 17$ $color(red)(a=-17)$

$use (5)$ $"use (5)"$

$- 17 + b = - 32$ $-17+b=-32$

$b = - 32 + 17$ $b=-32+17$

$b = - 15$ $color(red)(b=-15)$

$use (2)$ $"use (2)"$

$8 (- 17) - 15 + c = - 65$ $8(-17)-15+c=-65$

$- 136 - 15 + c = - 65$ $-136-15+c=-65$

$c = - 65 + 151$ $c=-65+151$

$c = 86$ $color(red)(c=86)$

$r:radius of circle$ $"r:radius of circle"$

$r = \frac{\sqrt{a^{2} + b^{2} - 4 c}}{2}$ $r=sqrt(a^2+b^2-4c)/2$

$r = \frac{\sqrt{289 + 225 - 344}}{2}$ $r=sqrt(289+225-344)/2$

$r = \sqrt{514 - 344}$ $r=sqrt(514-344)$

$r = \frac{\sqrt{170}}{2}$ $r=sqrt(170)/2$

$r^{2} = \frac{170}{4} = 42.5$ $r^2=170/4=42.5$

$a r e a = π \cdot r^{2}$ $area=pi*r^2$

$a r e a = 42.5 π$ $area=42.5pi$

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A triangle has corners at $(5, 2)$ $(5 ,2 )$ , $(8, 1)$ $(8 ,1 )$ , and $(3, 4)$ $(3 ,4 )$ . What is the area of the triangle's circumscribed circle?

1 Answer

Explanation:

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A triangle has corners at (5,2)(5 ,2 ), (8,1)(8 ,1 ), and (3,4)(3 ,4 ). What is the area of the triangle's circumscribed circle?

1 Answer

Explanation:

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A triangle has corners at $(5, 2)$ $(5 ,2 )$ , $(8, 1)$ $(8 ,1 )$ , and $(3, 4)$ $(3 ,4 )$ . What is the area of the triangle's circumscribed circle?