Math Problem Statement
Find a relationship between x and y such that (x,y) is equidistant from the two points. (4,-1),(-3,5)
Solution
To find a relationship between and such that the point is equidistant from the two given points and , we can use the concept that the distance between two points in the plane is given by the distance formula.
Step 1: Distance Formula
The distance between two points and is given by:
Step 2: Apply the Distance Formula
Let be the point that is equidistant from and . This means the distance from to must equal the distance from to . Thus, we have the following equation:
Step 3: Eliminate the Square Roots
To eliminate the square roots, square both sides of the equation:
Step 4: Expand Both Sides
Now, expand both sides of the equation:
Step 5: Simplify the Equation
Cancel out the and terms (since they appear on both sides):
Combine like terms:
Step 6: Rearrange Terms
Move all terms involving and to one side:
Simplify:
Step 7: Final Equation
Divide the entire equation by 2 to simplify it:
Thus, the relationship between and such that the point is equidistant from the points and is:
This is the equation of the perpendicular bisector of the line segment joining the two points.
Would you like more details or clarification on any step?
Here are 5 related questions to explore:
- How do you find the midpoint between two points?
- What is the equation of a circle passing through points and ?
- How can you find the slope of the line segment joining two points?
- What is the geometric significance of the perpendicular bisector?
- How do you calculate the distance between two points in 3D space?
Tip: The perpendicular bisector is always equidistant from two points—useful for constructing geometric shapes like triangles.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Geometry
Coordinate Geometry
Perpendicular Bisectors
Formulas
Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Equation of a line: Ax + By = C
Theorems
Perpendicular Bisector Theorem
Suitable Grade Level
Grades 9-11
Related Recommendation
Find the Equation for Points Equidistant from (3, 1) and (2, 0)
Find Equation of Point Equidistant from (-2, 3) and (4, -1)
Finding the Relation Between x and y for Equidistant Points (3,6) and (-3,4)
Find the Equation of the Locus of Equidistant Points: (x, y)
Find a Point on the y-axis Equidistant from Points A(6, 5) and B(-4, 3)