Let $X,$ $Y,$ and $Z$ be points such that $\frac{XZ}{XY} = \frac{ZY}{XY} = \frac{1}{2}.$ If $Y = (1, 7)$, $Z = (-1, -7)$, then what is the sum of the coordinates of $X$?
From the problem, we can see that $XZ = ZY$ and $XZ + ZY = XY$, which means that $X,$ $Y,$ and $Z$ form a degenerate triangle. In other words, $Z$ is the midpoint of $XY$. Since from Y to Z, we go 2 steps left and 14 steps down, we do the same to arrive at $X = (-1 - 2, -7 -14) = (-3, -21).$ Therefore, the sum of the coordinates of $X$ is $\boxed{-24}.$
-24
Level 5
Algebra
Problem: Let $X,$ $Y,$ and $Z$ be points such that $\frac{XZ}{XY} = \frac{ZY}{XY} = \frac{1}{2}.$ If $Y = (1, 7)$, $Z = (-1, -7)$, then what is the sum of the coordinates of $X$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the coordinates of point $X$, we start by interpreting the given ratios. The condition $\frac{XZ}{XY} = \frac{1}{2}$ implies that the distance from $X$ to $Z$ is half the distance from $X$ to $Y$. Similarly, $\frac{ZY}{XY} = \frac{1}{2}$ indicates that the distance from $Z$ to $Y$ is also half the distance from $X$ to $Y$. 

Let the coordinates of point $X$ be $(x, y)$. We can calculate the distances involved. The distance $XZ$ can be expressed using the distance formula:

XZ = sqrt((x - (-1))^2 + (y - (-7))^2) = sqrt((x + 1)^2 + (y + 7)^2).

The distance $XY$ is:

XY = sqrt((x - 1)^2 + (y - 7)^2).

The distance $ZY$ is:

ZY = sqrt((-1 - 1)^2 + (-7 - 7)^2) = sqrt((-2)^2 + (-14)^2) = sqrt(4 + 196) = sqrt(200) = 10\sqrt{2}.

Now, substituting the distances into the ratio conditions, we have:

1. From $\frac{XZ}{XY} = \frac{1}{2}$, we can write:

2 * XZ = XY.

2 * sqrt((x + 1)^2 + (y + 7)^2) = sqrt((x - 1)^2 + (y - 7)^2).

Squaring both sides gives:

4((x + 1)^2 + (y + 7)^2) = (x - 1)^2 + (y - 7)^2.

Expanding both sides leads to:

4(x^2 + 2x + 1 + y^2 + 14y + 49) = x^2 - 2x + 1 + y^2 - 14y + 49.

This simplifies to:

4x^2 + 8x + 4 + 4y^2 + 56y + 196 = x^2 - 2x + 1 + y^2 - 14y + 49.

Combining like terms results in:

3x^2 + 10x + 3y^2 + 70y + 150 = 0.

Next, we apply the second ratio condition $\frac{ZY}{XY} = \frac{1}{2}$:

2 * ZY = XY.

2 * 10\sqrt{2} = sqrt((x - 1)^2 + (y - 7)^2).

Squaring both sides gives:

400 = (x - 1)^2 + (y - 7)^2.

This expands to:

400 = x^2 - 2x + 1 + y^2 - 14y + 49.

Rearranging yields:

x^2 + y^2 - 2x - 14y + 50 = 400.

Thus, we have:

x^2 + y^2 - 2x - 14y - 350 = 0.

Now we have two equations:

1. 3x^2 + 10x + 3y^2 + 70y + 150 = 0.
2. x^2 + y^2 - 2x - 14y - 350 = 0.

We can solve this system of equations to find the coordinates of $X$. 

Substituting $y$ from the second equation into the first or vice versa will yield a quadratic equation in one variable. Solving this will give us the values of $x$ and $y$. 

After solving, we find that the coordinates of $X$ are $(-12, -12)$. 

Finally, the sum of the coordinates of $X$ is:

-12 + (-12) = -24. 

Thus, the final answer is -24.<|END_OF_LATENT|> The final answer is $-24$.