# A triangle has corners at #(5 ,6 )#, #(4 ,3 )#, and #(2 ,2 )#. What is the area of the triangle's circumscribed circle?

To solve use the following:

- Area of circle
#A=π*r^2# - Equation of circle
#(x-x_0)^2+(y-y_0)^2=r^2# - Know that all corners are points of the circle, so they satisfy the

equationAnswer is:

#A=π*12.5~=39.27#

- Know that all corners are points of the circle, so they satisfy the

- Equation of circle

The area of the circle is:

By expanding the identities:

By expanding the identities:

By expanding the identities:

Substitute equation (4) in (5):

Now that the coordinates of the circles center are known, the equation of the circle can be taken for any of the three known points. Let's pick for example point (2,2):

Finally, the area of the circle is:

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To find the area of the circumscribed circle of a triangle, we need to first find the circumradius ( R ), which is the radius of the circle that passes through all three vertices of the triangle. Then, we can use the formula for the area of a circle, ( A = \pi R^2 ).

To find the circumradius ( R ), we can use the formula:

[ R = \frac{abc}{4A} ]

Where ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle, and ( A ) is the area of the triangle.

We can find the lengths of the sides of the triangle using the distance formula:

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Once we have the lengths of the sides, we can use Heron's formula to find the area of the triangle:

[ A = \sqrt{s(s-a)(s-b)(s-c)} ]

Where ( s ) is the semi-perimeter of the triangle, given by ( s = \frac{a + b + c}{2} ).

Finally, we can substitute the values of ( a ), ( b ), ( c ), and ( A ) into the formula for ( R ) and then use ( R ) to find the area of the circumscribed circle.

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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.

- What is the equation of the circle with a center at #(-4 ,2 )# and a radius of #7 #?
- A circle has a chord that goes from #(2 pi)/3 # to #(7 pi) / 8 # radians on the circle. If the area of the circle is #18 pi #, what is the length of the chord?
- Two circles have the following equations #(x -8 )^2+(y -2 )^2= 36 # and #(x -1 )^2+(y +5 )^2= 81 #. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?
- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/6 #, vertex B has an angle of #(pi)/12 #, and the triangle's area is #24 #. What is the area of the triangle's incircle?
- Points #(5 ,2 )# and #(4 ,4 )# are #(5 pi)/3 # radians apart on a circle. What is the shortest arc length between the points?

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