How to calculate the X distance while Z object is moving top to bottom?

Question

First of all, sorry for my extremely low knowledge about mathematics. All i am able to do is this image which describe the problem.

I want to know that how can i calculate the X distance each time while Z object is moving continuously from top to bottom.

Note: This is not an assignment.

score 0 · Accepted Answer · answered Apr 16 '16 at 14:34

Note that $$AB^2+BC^2=(18-6)^2+(16-10)^2=AC^2$$

So $\triangle ABC$ is a right-angled triangle.

Let the smaller triangle be $\triangle ADE$. It is obvious that $\triangle ADE\sim \triangle ABC$($\because AAA$).

Similar triangles have the same ratio for every pairs of sides. In this problem,

$$\frac{DE}{BC}=\frac{AZ}{AB}$$

Let $Z=(10,a)$, $6\le a\le 18$. We have

$$\frac{x}{6}=\frac{a-6}{12}$$

$$x=\frac{a-6}2$$

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