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# Coordinates

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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

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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:

$$\sqrt{104}$$

This simplifies to:

$$2\sqrt{26}$$

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