SOLUTION: hey, i need help for this :) Find the point (𝑥,𝑦) on the line 𝑦=𝑥 that is equidistant from the points (6,8) and (8,0). Can you show me the steps too please, Thank you.

Question 1165878: hey, i need help for this :) Find the point (𝑥,𝑦) on the line 𝑦=𝑥 that is equidistant from the points (6,8) and (8,0). Can you show me the steps too please, Thank you.

Found 2 solutions by josgarithmetic, KMST:
Answer by josgarithmetic(39616) (Show Source): You can put this solution on YOUR website!
Such point on the line y=x means that vertical and horizontal coordinates are equal. Distance Formula could help.

and x and y values on that line to be equal,

--------------and same coordinate value as the one for y.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
There's more that one way to do it.

CONSIDERING DISTANCES:

The square of the distance from a point (x,y) to the point (6,8) is

In the case the point has ,
the square of that distance to (6,8) would be
.
The square of the distance from a point (x,y) to the point (8,0) is
, but if the point has ,
the distance to (6,8) would be the square root of
.
If the point you are looking for is at the same distance from (6.8,qnd (8,0),
those distances are the same, and their squares are the same, so

The point we are looking for is

ANOTHER APPROACH:
The points equidistant from and
are on the bisector of segment AB.
That is the line perpendicular to AB through the midpoint of AB.
The midpoint of AB is
The slope of AB is
The slope of a line perpendicular to AB is
The equation of the line going through ,
with slope is ' The intersection of that line with is the point that we look for.
Besides , it must comply with ,
so to find the value of we solve .

or
The point we are looking for is

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