There are two circles ${\mathrm{C}}_1$, and ${\mathrm{C}}_2$ of radii $3$ and $8$ units respectively. The common internal tangent T, touches the circles at points P and Q respectively. The line joining the centers of the circles intersects T at X. The distance of X from the center of the smaller circle is 5 units. What is the length of the line segment PQ?

Triangle ABC is a right-angled triangle. D and E are midpoints of AB and BC respectively. Read the following statements.

(i) $\mathrm{AE}=19$

(ii) $\mathrm{CD}=22$

(iii) Angle B is right angle.

Which of the following statements would be sufficient to determine the length of AC?

The figure below has been obtained by folding a rectangle. The total area of the figure (as visible) is $144$square meters. Had the rectangle not been folded, the current overlapping part would have been a square. What would have been the total area of the original unfolded rectangle?

In the diagram below, $\mathrm{CD}=\mathrm{BF}=10$ units and $\angle \mathrm{CED}=\angle \mathrm{BAF}=30°.$ What would be the area of triangle AED? (Note: Diagram below may not be proportional to scale.).