A triangle has corners at #(9 ,5 )#, #(2 ,3 )#, and #(7 ,6 )#. What is the area of the triangle's circumscribed circle?
The standard Cartesian form for the equation of a circle is:
Expand the squares:
Subtract equation [3.1] from equation [2.1]:
Combine like terms:
Subtract equation [3.1] from equation [4.1]:
Combine like terms:
#72 - 10h -6k - 3/2(93 - 14h - 4k) = 0
#72 - 10h -6k - 279/2 + 21h + 6k = 0
Combine like terms:
Use equation [6] to find the value of k:
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Circum center (6.1364, 1.7727)
Area of circumcircle 58.5065
Slope of AB m1 = (3-5)/(2-9)=2/7. Slope of BC m2 = (6-3)/(7-2) = 3/5. Solving Eqns (1), (2) Area of circumcircle
Slope of perpendicular at mid point of AB = -1/m1 = -7/2
Midpoint of AB = (9+2)/2, (3+5)/2 = 11/2, 4
Eqn of perpendicular bisector of AB is
Slope of perpendicular at mid point of AB = -1/m1 = -5/3.
Midpoint of BC = (7+2)/2, (6+3)/2 = 9/2, 9/2
Eqn of perpendicular bisector of BC is
Circum center (6.1364, 1.7727)
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To find the area of the circumscribed circle of a triangle, you can use the formula:
( A = \frac{{abc}}{{4R}} )
Where ( A ) is the area of the circumscribed circle, ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle, and ( R ) is the radius of the circumscribed circle.
First, calculate the lengths of the sides of the triangle using the distance formula:
( d_{1} = \sqrt{{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}} )
( d_{2} = \sqrt{{(x_{3} - x_{2})^2 + (y_{3} - y_{2})^2}} )
( d_{3} = \sqrt{{(x_{3} - x_{1})^2 + (y_{3} - y_{1})^2}} )
Then, calculate the semiperimeter (( s )) of the triangle:
( s = \frac{{d_{1} + d_{2} + d_{3}}}{2} )
Next, use Heron's formula to find the area of the triangle:
( A_{triangle} = \sqrt{{s(s - d_{1})(s - d_{2})(s - d_{3})}} )
Once you have the area of the triangle, you can find the radius (( R )) of the circumscribed circle using the formula:
( R = \frac{{abc}}{{4A_{triangle}}} )
Then, use the formula for the area of the circumscribed circle:
( A = \pi R^2 )
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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