Find the coordinates of the point equidistant from point A(1, 2), B(3, -4), and C(5, -6)?

We will be using the distance formula for finding the coordinates of the point.

Answer: The coordinates of the point equidistant from point A(1, 2), B(3, -4), and C(5, -6) are P(11, 2).

Let's get the solution step by step.

Explanation:

The three given points are A(1, 2), B(3, -4), and C(5, -6).
 
Let P (x, y) be the point that is equidistant from the given three points.
 
So, PA = PB = PC ---- (1)

According to distance formula, we have distance between any two points (x, y) and (\(x_{1}\), \(y_{1}\)) = √(x - \(x_{1}\))² + (y - \(y_{1}\))².

Distance PA = √(x - 1)² + (y - 2)² 

Distance PB = √(x - 3)² + (y + 4)² 

Distance PC = √(x - 5)² + (y + 6)²

Substituting these in equation (1),

√(x - 1)² + (y - 2)²  = √(x - 3)² + (y + 4)² = √(x - 5)² + (y + 6)²

⇒ x² + 1 - 2x + y² + 4 – 4y = x² + 9 - 6x + y² + 16 + 8y = x² + 25 – 10x + y² + 36 + 12y

⇒ 1 – 2x + 4 – 4y = 9 - 6x + 16 + 8y = 25 – 10x + 36 + 12y 

Considering, 1 – 2x + 4 – 4y = 9 - 6x + 16 + 8y, we get,

⇒ 2x - 6y = 10  … (1)

Now, consider, 1 – 2x + 4 – 4y = 25 – 10x + 36 + 12y

⇒ x – 2y = 7  … (2)

Solving (1) and (2), we get x = 11, y = 2.

You can solve the system of equations using Cuemath's linear equation calculator.

Thus, the required point is (11, 2).

We can verify our answer by using the distance algebra calculator.

Hence, the coordinates of the point equidistant from point A(1, 2), B(3, -4), and C(5, -6) are P(11, 2).