1. **Stating the problem:** Calculate the length of the mid-segment (midline) MN of trapezoid ABCD, where M and N are midpoints of sides AB and CD respectively. 2. **Formula and rules:** The mid-segment (midline) of a trapezoid is parallel to the bases and its length is the average of the lengths of the two bases. Formula: $$MN = \frac{AB + CD}{2}$$ 3. **Given data:** From the figure, the segments inside the trapezoid are labeled 4 and 3 along MN, indicating that MN is divided into parts 4 and 3, so: $$MN = 4 + 3 = 7$$ 4. **Using the formula:** $$7 = \frac{AB + CD}{2}$$ Multiply both sides by 2: $$2 \times 7 = AB + CD$$ $$14 = AB + CD$$ 5. **Conclusion:** The sum of the lengths of the bases AB and CD is 14 units. --- **Problem 17:** 1. **Stating the problem:** Calculate the length of the mid-segment (midline) of an isosceles trapezoid KLMN, given the diagonal LM = 24 cm and angle $\angle MKN = 60^\circ$. 2. **Key properties:** - In an isosceles trapezoid, the diagonals are equal. - The mid-segment length is the average of the two bases. - The trapezoid KLMN has diagonal LM = 24 cm. 3. **Approach:** Use triangle KLM formed by points K, L, M. Given $\angle MKN = 60^\circ$, and diagonal LM = 24 cm. 4. **Using Law of Cosines in triangle KLM:** Since $\angle MKN = 60^\circ$, and LM = 24 cm, we can find the length of the bases or mid-segment. However, since the problem only asks for the mid-segment length, and the trapezoid is isosceles, the mid-segment length equals the projection of the diagonal onto the base direction. 5. **Calculate mid-segment length:** The mid-segment length is: $$MN = LM \times \cos 60^\circ = 24 \times \frac{1}{2} = 12$$ 6. **Conclusion:** The length of the mid-segment MN is 12 cm.