# A triangle has corners at #(5 ,1 )#, #(7 ,9 )#, and #(4 ,3 )#. What is the area of the triangle's circumscribed circle?

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To find the area of the circumscribed circle of a triangle given its vertices, you can use the circumradius formula ( R = \frac{abc}{4A} ), where ( a ), ( b ), and ( c ) are the side lengths of the triangle, and ( A ) is the area of the triangle.

First, calculate the side lengths of the triangle using the distance formula:

Side ( a ): [ a = \sqrt{(5-7)^2 + (1-9)^2} = \sqrt{4^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5} ]

Side ( b ): [ b = \sqrt{(7-4)^2 + (9-3)^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} ]

Side ( c ): [ c = \sqrt{(4-5)^2 + (3-1)^2} = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} ]

Next, calculate the area of the triangle using Heron's formula: [ s = \frac{a + b + c}{2} = \frac{4\sqrt{5} + 3\sqrt{5} + \sqrt{5}}{2} = \frac{8\sqrt{5}}{2} = 4\sqrt{5} ] [ A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{4\sqrt{5}(4\sqrt{5}-4)(4\sqrt{5}-3)(4\sqrt{5}-\sqrt{5})} ] [ A = \sqrt{4\sqrt{5}(4\sqrt{5}-4)(4\sqrt{5}-3)(3\sqrt{5})} = \sqrt{4 \cdot 4 \cdot 3 \cdot 5} = \sqrt{240} = 4\sqrt{15} ]

Now, use the circumradius formula to find the circumradius ( R ): [ R = \frac{abc}{4A} = \frac{(4\sqrt{5})(3\sqrt{5})(\sqrt{5})}{4(4\sqrt{15})} = \frac{60\sqrt{5}}{16\sqrt{15}} = \frac{15}{4} = 3.75 ]

The area of the circumscribed circle is given by ( \pi R^2 ): [ \text{Area} = \pi (3.75)^2 = \pi (14.0625) \approx 44.15 ]

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To find the area of the circumscribed circle of a triangle, we can use the formula:

Area = (abc) / (4R)

Where: a, b, and c are the lengths of the sides of the triangle, and R is the radius of the circumscribed circle.

First, we need to find the lengths of the sides of the triangle using the distance formula:

Distance = √[(x2 - x1)² + (y2 - y1)²]

Then, we can calculate the semiperimeter (s) of the triangle:

s = (a + b + c) / 2

Next, we use Heron's formula to find the area of the triangle:

Area = √[s(s - a)(s - b)(s - c)]

After finding the area of the triangle, we can find the radius of the circumscribed circle (R) using the formula:

R = (abc) / (4Area)

Finally, we can substitute the values into the formula 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.

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