[Math] How to compute the length of line segments inside an isosceles trapezoid parallel to the base

Question

here is the problem in a picture, sorry for the drawing. i have found how to compute the midline of the isosceles trapezoid, 1/2 (top+bottom) , but there seems to be no formula for other line segments inside aside from the midline. I need to know the length of the x and y values in the photo. Thank you

Answer 1

Suppose the top line is $ \overline{AB}$ and the bottom line is $\overline{CD}$. Now, find a point $E$ in $\overline{CD}$ such that $\overline{AE}$ is parallel to $\overline{BD}$. Then apply the rule for the ratio of similar triangles to the triangle $\overline{ACE}$.

Update: $$\frac{AB}{AD}=\frac{AC}{AE}=\frac{BC}{DE}$$ if $\triangle ABC$ and $\triangle ADE$ are similar triangles.

Update 2: I forget to tell you another fact. If $\triangle ABC$ and $\triangle ADE$ are similar triangles, $$\frac{\text{height of }\triangle ABC}{\text{height of } \triangle ADE}=\frac{AB}{AD}$$