SOLUTION: A(-2,4) and B(-5,-3) are two points, find a point P on the y-axis, such that PA = PB. Please help!! Thanks

Question 784515: A(-2,4) and B(-5,-3) are two points, find a point P on the y-axis, such that PA = PB.
Please help!!

Thanks
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
A point on the y-axis has .
It's the point P(0,y).
We just need to find the coordinate.

The square of a distance can be calculated as the square of the distance along the x-direction plus the square of the distance along the x-direction.
(That's the Pythagorean relation).
Calculating the distance would require calculating the square root of that sum, but we will not worry about that for now.

The square of the distance between P(0,y) and A(-2,4) is

The square of the distance between P(0,y) and B(-5,-3) is

If the distances are the same, the squares of the distances will be equal, so the solution to the problem (if any) must be among the solutions to

We found only one solution, so we should just verify that it works.
I can count the horizontal and vertical distance on the graph.
I could calculate distances by squaring, adding and taking square roots, but it is not needed and does not help to understand anything.
The horizontal and vertical distances betweem A and P are 2 and 5 respectively.
The horizontal and vertical distances betweem B and P are 5 and 2 respectively.
That tells me that the distances AP and BP are the same.
They are hypotenuses of right triangles with the same leg lengths (congruent right triangles), so have the same length.
Why would I need lengthy explanations and/or ugly formulas/calculations if my graph is worth a thousand words.
If you are lucky, you will be able to understand, and will only need to just show the solution, and at most a streamlined (few steps) version of the solving of the equation.

Unfortunately, teachers sometimes like complicated, unnecessary calculations, and even ugly looking formulas, so here they go.
The horizontal and vertical distances betweem A and P are:
and
The distance between A and P is

The horizontal and vertical distances betweem B and P are:
and
The distance between B and P is

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