$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow ZXY$. If $AB = 12, BC = 8, CA = 10$ and $ZX = 10$,then $XY + YZ = \ldots$

$D$ is a point on the side $BC$ of a triangle $ABC$ such that $\angle ADC = \angle BAC$. Show that $CA^2 = CB \cdot CD$.

In quadrilateral $ABCD$,$\overline{AB} \parallel \overline{CD}$ and $\overline{AC} \cap \overline{BD} = \{O\}$. If $OA = 3x - 19$,$OB = x - 4$,$OC = x - 3$,and $OD = 4$,find the value of $x$.

$\Delta ABC$ and $\Delta PQR$ are equilateral triangles. If $\frac{AB}{PQ} = \frac{3}{2}$,prove that $4 \times \text{Area of } \Delta ABC = 9 \times \text{Area of } \Delta PQR$.

In $\square ABCD$,$\overline{AB} \parallel \overline{CD}$ and $\overline{AC} \cap \overline{BD} = \{M\}$. If $MA = 6$,$MB = 9$ and $MC = 8$,then $MD = \dots$

The areas of two similar triangles are $200$ and $128$. Then,the ratio of their corresponding altitudes is.....