Find the equation of a circle whose vertices are A (2, 3), B(5, 4), C(3, 7).

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Find the equation of the circle in the triangle whose varticles are A(2,3) B(5;4) and c(5,7)

To find the equation of a circle given the coordinates of three distinct points on its circumference, you can follow these steps:

Step 1: Find the equation of the perpendicular bisectors of two of the line segments formed by the three given points.

Step 2: Determine the intersection point of these two perpendicular bisectors. This point will be the center of the circle.

Step 3: Calculate the distance between this center point and one of the given points. This distance will be the radius of the circle.

Now let's go step by step:

Step 1: Find the equations of the perpendicular bisectors.

Let's start with points A(2, 3) and B(5, 4). The midpoint of segment AB can be found by using the midpoint formula:
Midpoint = [(x₁ + x₂) / 2, (y₁ + y₂) / 2]

Midpoint of AB = [(2 + 5)/2, (3 + 4)/2]
= [7/2, 7/2]
= (7/2, 7/2)

The slope of AB can be found using the slope formula:
Slope = (y₂ - y₁) / (x₂ - x₁)

Slope of AB = (4 - 3) / (5 - 2)
= 1/3

Since the perpendicular bisector of AB will have a negative reciprocal slope, the slope of the perpendicular bisector, let's call it m₁, will be -3 (negative reciprocal of 1/3).

Using the equation y = mx + b, where m₁ = -3 and (x, y) = (7/2, 7/2), we can solve for the y-intercept, b₁:

7/2 = -3(7/2) + b₁
7/2 = -21/2 + b₁
7/2 + 21/2 = b₁
28/2 = b₁
14 = b₁

Therefore, the equation of the perpendicular bisector of AB is:
y = -3x + 14

Now, let's find the equation of the perpendicular bisector of BC.

Using the same approach, we find that the midpoint of segment BC is:
Midpoint of BC = (4, 5.5)

The slope of BC is:
Slope of BC = (7 - 4) / (3 - 5)
= 3/(-2)
= -3/2

The slope of the perpendicular bisector, m₂, will be the negative reciprocal of -3/2:
m₂ = -2/3

Using the equation y = mx + b, and substituting (x, y) = (4, 5.5), we can solve for b₂:

5.5 = (-2/3)(4) + b₂
5.5 = -8/3 + b₂
5.5 + 8/3 = b₂
(16.5 + 8)/3 = b₂
24.5/3 = b₂

Therefore, the equation of the perpendicular bisector of BC is:
y = (-2/3)x + 24.5/3

Step 2: Determine the intersection point of the perpendicular bisectors.

To find the center of the circle, we need to find the intersection point of the two perpendicular bisectors.

Solving the two equations
y = -3x + 14
y = (-2/3)x + 24.5/3

we get:
-3x + 14 = (-2/3)x + 24.5/3

Multiply the entire equation by 3 to get rid of the fractions:
-9x + 42 = -2x + 24.5

Simplify the equation:
-9x + 2x = 24.5 - 42
-7x = -17.5

Divide both sides by -7:
x = -17.5 / -7
x = 2.5

Substituting the value of x into one of the equations (let’s use the first one), we can find y:
y = -3(2.5) + 14
y = -7.5 + 14
y = 6.5

Therefore, the intersection point (center of the circle) is (2.5, 6.5).

Step 3: Calculate the radius of the circle.

To determine the radius, we need the distance between the center of the circle and one of the given points. Let's use point A(2, 3).

Using the distance formula:
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Distance = √[(2.5 - 2)² + (6.5 - 3)²]
= √[(0.5)² + (3.5)²]
= √[0.25 + 12.25]
= √12.5

Therefore, the radius of the circle is √12.5.

In conclusion, the equation of the circle with vertices A(2, 3), B(5, 4), C(3, 7) is:
(x - 2.5)² + (y - 6.5)² = 12.5