1. **State the problem:** We need to determine whether the distance between the tips of the clock hands is greater at 4:00 or 7:00 using the Hinge Theorem. 2. **Recall the Hinge Theorem:** It states that if two triangles have two pairs of equal sides, then the triangle with the larger included angle has the longer third side. 3. **Apply to clock hands:** The lengths of the hour and minute hands are constant, so the distance between tips depends on the angle between them. 4. **Calculate the angle at 4:00:** The hour hand at 4 is at $4 \times 30^\circ = 120^\circ$ from 12 o'clock. The minute hand at 12 is at $0^\circ$. The angle between them is $|120^\circ - 0^\circ| = 120^\circ$. 5. **Calculate the angle at 7:00:** The hour hand at 7 is at $7 \times 30^\circ = 210^\circ$. The minute hand at 12 is at $0^\circ$. The angle between them is $|210^\circ - 0^\circ| = 210^\circ$. However, the smaller angle between hands is $360^\circ - 210^\circ = 150^\circ$. 6. **Compare angles:** The angle between hands at 4:00 is $120^\circ$, and at 7:00 is $150^\circ$. 7. **Conclusion:** Since the angle at 7:00 ($150^\circ$) is larger than at 4:00 ($120^\circ$), by the Hinge Theorem, the distance between the tips of the hands is greater at 7:00. 8. **How distance changes throughout the day:** The angle between the hands changes continuously as the hour hand moves $0.5^\circ$ per minute and the minute hand moves $6^\circ$ per minute, causing the distance between tips to vary periodically. **Final answer:** The distance between the tips of the hands is greater at 7:00 because the angle between the hands is larger at 7:00 than at 4:00.