Let $ABCDEFGHIJ$ be a regular $10$-sided polygon that has all its vertices in one circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at $P$ and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the measure of the  segment $PQ$.
1. **Determine the internal angle of the regular 10-sided polygon:**
   Since $ABCDEFGHIJ$ is a regular 10-sided polygon, each internal angle can be calculated as:
   \[
   \text{Internal angle} = 180^\circ - \frac{360^\circ}{10} = 180^\circ - 36^\circ = 144^\circ
   \]

2. **Analyze the isosceles trapezium $AJIH$:**
   Since $\angle AJI = \angle JIH$ and $AJ = IH$, $AJIH$ is an isosceles trapezium. The angles at the base of the trapezium can be calculated as:
   \[
   \angle IHA = \angle JAH = \frac{360^\circ - 2 \cdot 144^\circ}{2} = \frac{360^\circ - 288^\circ}{2} = \frac{72^\circ}{2} = 36^\circ
   \]

3. **Analyze the quadrilateral $BAJI$:**
   Similarly, in the quadrilateral $BAJI$, we get the same result since both quadrilaterals are congruent.

4. **Determine the properties of $APIJ$:**
   Since $AJ = JI$ and $APIJ$ is a parallelogram, $APIJ$ is a rhombus. Similarly, $BCDQ$ is also a rhombus.

5. **Analyze the isosceles trapezium $PABQ$:**
   Since $\angle BAP = \angle ABQ = 108^\circ$ and $PA = AB = BQ$, $PABQ$ is an isosceles trapezium.

6. **Calculate the length of $AB$:**
   Using the cosine rule in the isosceles trapezium:
   \[
   \frac{\frac{AB}{2}}{5} = \cos(180^\circ - 90^\circ - 18^\circ) = \cos(72^\circ)
   \]
   Therefore,
   \[
   AB = 10 \cdot \cos(72^\circ)
   \]

7. **Calculate $PQ$ in $PABQ$:**
   Using the properties of the isosceles trapezium:
   \[
   PQ = 10 \cos(72^\circ) + 20 \cos^2(72^\circ)
   \]
   We know that:
   \[
   \cos(72^\circ) = \frac{\sqrt{5} - 1}{4}
   \]
   Therefore,
   \[
   PQ = 10 \cdot \frac{\sqrt{5} - 1}{4} + 20 \cdot \left(\frac{\sqrt{5} - 1}{4}\right)^2
   \]
   Simplifying further:
   \[
   PQ = \frac{10(\sqrt{5} - 1)}{4} + 20 \cdot \frac{(\sqrt{5} - 1)^2}{16}
   \]
   \[
   PQ = \frac{10\sqrt{5} - 10}{4} + \frac{5(5 - 2\sqrt{5} + 1)}{4}
   \]
   \[
   PQ = \frac{10\sqrt{5} - 10 + 25 - 10\sqrt{5} + 5}{4}
   \]
   \[
   PQ = \frac{20}{4}
   \]
   \[
   PQ = 5
   \]

The final answer is $\boxed{PQ = 5}$