Understanding Equidistance

Oct 13, 2025 by recoveryshake.com 84 views

===========================================================

Introduction

In geometry, the concept of equidistance is crucial in determining the points that are at the same distance from a given set of points. This concept is widely used in various fields, including mathematics, physics, and engineering. In this article, we will explore the problem of finding the coordinates of the point that is equidistant from three given points.

Problem Statement

Given the points ${$ (2,3)$}$, ${$ (0,-1)$}$, and ${$ (4,5)$$}$, we need to find the coordinates of the point that is equidistant from these three points.

Understanding Equidistance

Equidistance is a concept that refers to the distance between two or more points. In this case, we are looking for a point that is at the same distance from the three given points. This means that the distance between the point and each of the three given points is equal.

Mathematical Formulation

Let's denote the coordinates of the point we are looking for as [ $(x,y)\$\$. The distance between two points \[$ (x_1,y_1)$$ and [$(x_2,y_2)$$ is given by the formula:

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

We can use this formula to calculate the distance between the point [$(x,y)$$ and each of the three given points.

Calculating Distances

The distance between the point [ $(x,y)\$\$ and the point \[$ (2,3)$$ is given by:

d_1 = \sqrt{(x-2)^2 + (y-3)^2}

The distance between the point [ $(x,y)\$\$ and the point \[$ (0,-1)$$ is given by:

d_2 = \sqrt{(x-0)^2 + (y-(-1))^2}

The distance between the point [ $(x,y)\$\$ and the point \[$ (4,5)$$ is given by:

d_3 = \sqrt{(x-4)^2 + (y-5)^2}

Equating Distances

Since the point we are looking for is equidistant from the three given points, we can set up the following equations:

d_1 = d_2

d_1 = d_3

Substituting the expressions for $d_1$ , $d_2$ , and $d_3$ , we get:

\sqrt{(x-2)^2 + (y-3)^2} = \sqrt{(x-0)^2 + (y-(-1))^2}

\sqrt{(x-2)^2 + (y-3)^2} = \sqrt{(x-4)^2 + (y-5)^2}

Simplifying Equations

Squaring both sides of each equation, we get:

(x-2)^2 + (y-3)^2 = (x-0)^2 + (y-(-1))^2

(x-2)^2 + (y-3)^2 = (x-4)^2 + (y-5)^2

Expanding and simplifying the equations, we get:

x^2 - 4x + 4 + y^2 - 6y + 9 = x^2 + y^2 + 1

x^2 - 4x + 4 + y^2 - 6y + 9 = x^2 - 8x + 16 + y^2 - 10y + 25

Solving Equations

Simplifying the equations further, we get:

-4x - 6y + 13 = 1

-4x - 6y + 13 = -8x - 10y + 41

Subtracting the first equation from the second equation, we get:

0 = -4x - 4y + 28

Simplifying the equation, we get:

x + y = 7

Finding the Point

We have found that the point we are looking for lies on the line $x + y = 7$ . To find the coordinates of the point, we need to find the intersection of this line with the perpendicular bisectors of the segments joining the point to each of the three given points.

Perpendicular Bisectors

The perpendicular bisector of a segment is a line that passes through the midpoint of the segment and is perpendicular to the segment. We can find the equations of the perpendicular bisectors of the segments joining the point to each of the three given points.

Midpoints

The midpoint of the segment joining the point [ $(x,y)\$\$ and the point \[$ (2,3)$$ is given by:

\left(\frac{x+2}{2}, \frac{y+3}{2}\right)

The midpoint of the segment joining the point [ $(x,y)\$\$ and the point \[$ (0,-1)$$ is given by:

\left(\frac{x+0}{2}, \frac{y-1}{2}\right)

The midpoint of the segment joining the point [ $(x,y)\$\$ and the point \[$ (4,5)$$ is given by:

\left(\frac{x+4}{2}, \frac{y+5}{2}\right)

Slopes

The slope of the segment joining the point [ $(x,y)\$\$ and the point \[$ (2,3)$$ is given by:

m_1 = \frac{3-y}{2-x}

The slope of the segment joining the point [ $(x,y)\$\$ and the point \[$ (0,-1)$$ is given by:

m_2 = \frac{-1-y}{x-0}

The slope of the segment joining the point [ $(x,y)\$\$ and the point \[$ (4,5)$$ is given by:

m_3 = \frac{5-y}{x-4}

Perpendicular Bisectors

The equation of the perpendicular bisector of the segment joining the point [ $(x,y)\$\$ and the point \[$ (2,3)$$ is given by:

y - \frac{y+3}{2} = -\frac{1}{m_1}\left(x - \frac{x+2}{2}\right)

The equation of the perpendicular bisector of the segment joining the point [ $(x,y)\$\$ and the point \[$ (0,-1)$$ is given by:

y - \frac{y-1}{2} = -\frac{1}{m_2}\left(x - \frac{x}{2}\right)

The equation of the perpendicular bisector of the segment joining the point [ $(x,y)\$\$ and the point \[$ (4,5)$$ is given by:

y - \frac{y+5}{2} = -\frac{1}{m_3}\left(x - \frac{x+4}{2}\right)

Finding the Intersection

We can find the intersection of the line $x + y = 7$ with each of the perpendicular bisectors.

Intersection with Perpendicular Bisector 1

Solving the system of equations:

x + y = 7

y - \frac{y+3}{2} = -\frac{1}{m_1}\left(x - \frac{x+2}{2}\right)

we get:

x = 1

y = 6

Intersection with Perpendicular Bisector 2

Solving the system of equations:

x + y = 7

y - \frac{y-1}{2} = -\frac{1}{m_2}\left(x - \frac{x}{2}\right)

we get:

x = 1

y = 6

Intersection with Perpendicular Bisector 3

Solving the system of equations:

x + y = 7

y - \frac{y+5}{2} = -\frac{1}{m_3}\left(x - \frac{x+4}{2}\right)

we get:

x = 1

y = 6

Conclusion

We have found that the point [ $(1,6)\$\$ is equidistant from the points \[$ (2,3)$$, [ $(0,-1)\$\$, and \[$ (4,5)$$. This point lies on the line $x + y = 7$ and is the intersection of this line with the perpendicular bisectors of the segments joining the point to each of the three given points.

Final Answer

The final answer is [$(1,6)$$.

===========================================================

Introduction

In our previous article, we explored the problem of finding the coordinates of the point that is equidistant from three given points. We used the concept of equidistance and the formula for distance between two points to find the point that lies on the line $x + y = 7$ and is the intersection of this line with the perpendicular bisectors of the segments joining the point to each of the three given points.

Q&A

Q: What is the concept of equidistance?

A: Equidistance is a concept that refers to the distance between two or more points. In this case, we are looking for a point that is at the same distance from the three given points.

Q: How do we find the point that is equidistant from three given points?

A: We can use the formula for distance between two points to calculate the distance between the point and each of the three given points. We then set up equations to equate the distances and solve for the coordinates of the point.

Q: What is the formula for distance between two points?

A: The formula for distance between two points [ $(x_1,y_1)\$\$ and \[$ (x_2,y_2)$$ is given by:

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Q: How do we find the perpendicular bisectors of the segments joining the point to each of the three given points?

A: We can find the midpoints of the segments and the slopes of the segments. We then use the midpoint and slope to find the equation of the perpendicular bisector.

Q: What is the significance of the line $x + y = 7$ in this problem?

A: The line $x + y = 7$ is the locus of points that are equidistant from the three given points. The point we are looking for lies on this line.

Q: How do we find the intersection of the line $x + y = 7$ with the perpendicular bisectors of the segments joining the point to each of the three given points?

A: We can solve the system of equations formed by the line and the perpendicular bisector to find the coordinates of the point.

Q: What is the final answer to this problem?

A: The final answer is [$(1,6)$$.

Additional Tips and Tricks

Make sure to use the correct formula for distance between two points.
Use the midpoint and slope to find the equation of the perpendicular bisector.
Solve the system of equations formed by the line and the perpendicular bisector to find the coordinates of the point.
Check your work by plugging the coordinates back into the original equations.

Conclusion

Finding the coordinates of the point that is equidistant from three given points requires a thorough understanding of the concept of equidistance and the formula for distance between two points. By following the steps outlined in this article, you can find the point that lies on the line $x + y = 7$ and is the intersection of this line with the perpendicular bisectors of the segments joining the point to each of the three given points.

Final Answer

The final answer is [$(1,6)$$.