The distance between the points (x,21) and (4,7) is 10sqrt(3) Find the sum of all possible values of x.

 Jan 23, 2022

The first step is to consider some triangles.

Once we find x, we can see that there are two congruent triangles with hypotenuse length \(10\sqrt{3}\).

One of the legs of this triangle will be y1 - y2, which is 14.

Using the pythagorean theorem, we can find the distance between the two x coordinates. 

\(\sqrt{(10\sqrt{3})^2 - 14^2}\)

This simplifies to:


This simplifies to:



This means the x coordinate is \(2\sqrt{26}\) units away from the x coordinate of (4,7), and then we have these 2 possible values of x:

\(4 + 2\sqrt{26}\) and \(4 - 2\sqrt{26}\)


Since the question is looking for the sum of the possible values of x, then we just add the two possible values together:

\(4 + 2\sqrt{26} + 4 - 2\sqrt{26}\)

This simplifies to 8.


Therefore, the sum of all possible values of x is 8.




 Jan 23, 2022

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