If a point A (0,2) is equidistant from the points B (3,p) and C (p,5), then find the values of p.

To solve this problem, we need to use the concept of the distance formula. The distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula:

Distance = √((x₂ - x₁)² + (y₂ - y₁)²)

We are given that point A (0, 2) is equidistant from points B (3, p) and C (p, 5). This means the distance from A to B is equal to the distance from A to C. We will set up two distance equations and solve for p.

Step 1: Distance from A to B The distance from point A (0, 2) to point B (3, p) is: Distance AB = √((3 - 0)² + (p - 2)²) Distance AB = √(9 + (p - 2)²) Distance AB = √(9 + (p² - 4p + 4)) Distance AB = √(p² - 4p + 13)

Step 2: Distance from A to C The distance from point A (0, 2) to point C (p, 5) is: Distance AC = √((p - 0)² + (5 - 2)²) Distance AC = √(p² + 9) Distance AC = √(p² + 9)

Step 3: Set the distances equal to each other Since A is equidistant from B and C, we set the two distances equal to each other: √(p² - 4p + 13) = √(p² + 9)

Step 4: Square both sides to eliminate the square roots (p² - 4p + 13) = (p² + 9)

Step 5: Simplify the equation p² - 4p + 13 = p² + 9

Cancel out p² from both sides: -4p + 13 = 9

Step 6: Solve for p -4p = 9 - 13 -4p = -4 p = 1

Thus, the value of p is 1.