To find the radius of the circumcircle, we need to use the formula:
\[ R = \frac{abc}{4\sqrt{s(s-a)(s-b)(s-c)}} \]
where \( a, b, \) and \( c \) are the side lengths of the triangle, and \( s \) is the semi-perimeter (\( s = \frac{a+b+c}{2} \)).
First, calculate the side lengths using the given points A(1, 3), B(-3, 5), and C(5, -1):
\[ AB = \sqrt{(1 - (-3))^2 + (3 - 5)^2} \]
\[ BC = \sqrt{(5 - (-3))^2 + ((-1) - 5)^2} \]
\[ CA = \sqrt{(5 - 1)^2 + ((-1) - 3)^2} \]
Next, find the semi-perimeter \( s \), and then use the formula for the circumradius to find \( R \). Once you have the circumradius, you can use the distance formula to find \( AP \), where \( P \) is equidistant from A, B, and C.
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Step-by-step explanation:
To find the radius of the circumcircle, we need to use the formula:
\[ R = \frac{abc}{4\sqrt{s(s-a)(s-b)(s-c)}} \]
where \( a, b, \) and \( c \) are the side lengths of the triangle, and \( s \) is the semi-perimeter (\( s = \frac{a+b+c}{2} \)).
First, calculate the side lengths using the given points A(1, 3), B(-3, 5), and C(5, -1):
\[ AB = \sqrt{(1 - (-3))^2 + (3 - 5)^2} \]
\[ BC = \sqrt{(5 - (-3))^2 + ((-1) - 5)^2} \]
\[ CA = \sqrt{(5 - 1)^2 + ((-1) - 3)^2} \]
Next, find the semi-perimeter \( s \), and then use the formula for the circumradius to find \( R \). Once you have the circumradius, you can use the distance formula to find \( AP \), where \( P \) is equidistant from A, B, and C.