A triangle has corners at $(5, 1)$ $(5 ,1 )$ , $(2, 7)$ $(2 ,7 )$ , and $(7, 3)$ $(7 ,3 )$ . What is the area of the triangle's circumscribed circle?

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A triangle has corners at $(5, 1)$ $(5 ,1 )$ , $(2, 7)$ $(2 ,7 )$ , and $(7, 3)$ $(7 ,3 )$ . What is the area of the triangle's circumscribed circle?

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Answer 1 · 2018-06-10T12:00:29

Area of triangle's circumscribed circle is $35.78$ $35.78$ sq.unit.

Explanation:

The three corners are $A (5, 1) B (2, 7) and C (7, 3)$ $A (5,1) B (2,7) and C (7,3)$

Distance between two points $(x_{1}, y_{1}) and (x_{2}, y_{2})$ $(x_1,y_1) and (x_2,y_2)$ is

$D = \sqrt{{(x_{1} - x_{2})}_{1}^{2} + {(y_{1} - y_{2})}_{1}^{2}}$ $D= sqrt ((x_1-x_2)^2+(y_1-y_2)^2$

Side $A B = \sqrt{{(5 - 2)}^{2} + {(1 - 7)}^{2}} \approx 6.708$ $AB= sqrt ((5-2)^2+(1-7)^2) ~~ 6.708$ unit

Side $B C = \sqrt{{(2 - 7)}^{2} + {(7 - 3)}^{2}} \approx 6.403$ $BC= sqrt ((2-7)^2+(7-3)^2) ~~6.403$ unit

Side $C A = \sqrt{{(7 - 5)}^{2} + {(3 - 1)}^{2}} \approx 2.828$ $CA= sqrt ((7-5)^2+(3-1)^2) ~~ 2.828$ unit

Area of Triangle is

$A_{t} = ∣ ∣ ∣ \frac{1}{2} (x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})) ∣ ∣ ∣$ $A_t = |1/2(x_1(y_2−y_3)+x_2(y_3−y_1)+x_3(y_1−y_2))|$

$A_{t} = ∣ ∣ ∣ \frac{1}{2} (5 (7 - 3) + 2 (3 - 1) + 7 (1 - 7)) ∣ ∣ ∣$ $A_t = |1/2(5(7−3)+2(3−1)+7(1−7))|$ or

$A_{t} = ∣ ∣ ∣ \frac{1}{2} (20 + 4 - 42) ∣ ∣ ∣ = | - 9.0 | = 9.0$ $A_t = |1/2(20+4-42)| = | -9.0| =9.0$ sq.unit.

Radius of circumscribed circle is $R = \frac{A B \cdot B C \cdot C A}{4 \cdot A_{t}}$ $R=(AB*BC*CA)/(4*A_t)$ or

$R = \frac{\sqrt{45} \cdot \sqrt{41} \cdot \sqrt{9}}{4 \cdot 9} \approx 3.375$ $R=(sqrt(45)*sqrt(41)*sqrt(9))/(4*9) ~~ 3.375$ unit

Area of circumscribed circle is $A_{c} = π \cdot R^{2} = π \cdot {3.375}^{2} \approx 35.78$ $A_c=pi*R^2=pi*3.375^2~~35.78$

sq.unit [Ans]

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A triangle has corners at $(5, 1)$ $(5 ,1 )$ , $(2, 7)$ $(2 ,7 )$ , and $(7, 3)$ $(7 ,3 )$ . What is the area of the triangle's circumscribed circle?

1 Answer

Explanation:

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A triangle has corners at (5,1)(5 ,1 ), (2,7)(2 ,7 ), and (7,3)(7 ,3 ). What is the area of the triangle's circumscribed circle?

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A triangle has corners at $(5, 1)$ $(5 ,1 )$ , $(2, 7)$ $(2 ,7 )$ , and $(7, 3)$ $(7 ,3 )$ . What is the area of the triangle's circumscribed circle?