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### Finding a Point Equidistant From Two Other Points

```
Date: 8/18/96 at 17:5:44
From: Anonymous
Subject: Finding a Point Equidistant From Two Other Points

Point A is (-5,-3), and point B is (-1,-5). What should the value
of k be for the point (3, k) to be equidistant from A and B?

I tried to use the distance formula for this problem but I couldn't
solve for k that way.  What should I do?

A very confused Jason.
```

```
Date: 8/30/96 at 15:26:47
From: Doctor Jerry
Subject: Re: Finding a Point Equidistant From Two Other Points

If the distance from (3,k) to A and B is equal, then the squares of
the distances from (3,k) to A and B are also equal.  This will
simplify the calculations.

From (3,k) to (-5,-3), the square of the distance is 64+(k+3)^2.
From (3,k) to (-1,-5), the square of the distance is 16+(k+5)^2.

Setting these expressions equal,   64+(k+3)^2 = 16 +(k+5)^2

squaring,  64+k^2+6k+9 = 16+k^2+10k+25

simplifying,           4k = 32

So, k = 8.

Now check the squared distances.

From (3,8) to (-5,-3) is 64+121 = 185.
From (3,8) to (-1,-5) is 16+169 = 185.

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Coordinate Plane Geometry
High School Equations, Graphs, Translations
High School Geometry

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