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Equidistant Point

An equidistant point is a point that is located at the exact same distance from two or more given points. In coordinate geometry, if a point P is equidistant from points A and B, the distance from P to A is mathematically equal to the distance from P to B.

Key Concepts

To solve problems involving an equidistant point, we equate the distances from the unknown point to the given points using the distance formula.
Squaring both sides of the distance equation removes the square root, making it much easier to solve the resulting algebraic expression.
An equidistant point from two given points does not have to lie on the line segment connecting them; it can lie anywhere on the perpendicular bisector of that segment.

Formula & Equation

PA2=PB2,whichexpandsto(xāˆ’x1)2+(yāˆ’y1)2=(xāˆ’x2)2+(yāˆ’y2)2PA^2 = PB^2, which expands to (x - x1)^2 + (y - y1)^2 = (x - x2)^2 + (y - y2)^2

P(x, y) is the equidistant point, while A(x1, y1) and B(x2, y2) are the two given points.

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Common Misconceptions

Myth: An equidistant point from two points is always their exact midpoint.
Fact: While the midpoint is equidistant, it is not the only one. Any point on the perpendicular bisector of the line segment joining the two points is also equidistant from them.

Real World Applications

Determining the optimal location for a new fire station so it is exactly the same distance from two neighboring towns.
Programming a delivery drone to drop a package at a coordinate exactly halfway between two warehouses.

Frequently Asked Questions

How do you find a point on the x-axis that is equidistant from two given points?
Let the point on the x-axis be (x, 0). Use the distance formula to equate its distance from the two given points, square both sides to remove the roots, and solve for x.
Does equidistant mean parallel in coordinate geometry?
No, equidistant means equal in distance. It refers to a specific point having the exact same distance from two or more other points, not the angle or direction of lines.

Keep Learning

Distance FormulaMid-point FormulaCartesian Coordinate System