Problem Setup:

• We are given a parallelogram ABCD.
• P is the midpoint of the side BC.
• The line segment AP bisects the angle BAD, meaning $\angle BAP = \angle DAP$.
• The length of side AD is given as $10 \text{ cm}$.
• We need to find the length of side CD.

Parallelogram Properties and Angle Bisector Analysis

In a parallelogram ABCD, we know that opposite sides are equal in length and parallel.

• $AD = BC$ and $AB = CD$.
• $AD \parallel BC$.

Since $AD = 10 \text{ cm}$, we have $BC = 10 \text{ cm}$.

Analyzing Triangle ABP

We are given that AP bisects $\angle BAD$, so $\angle BAP = \angle DAP$.

Because $AD \parallel BC$, we can identify alternate interior angles formed by the transversal AP:

• $\angle DAP = \angle APB$.

Combining the angle equalities, we get:

• $\angle BAP = \angle DAP = \angle APB$.
• This implies $\angle BAP = \angle APB$.

In triangle ABP, two angles are equal ($\angle BAP = \angle APB$). Therefore, triangle ABP is an isosceles triangle, and the sides opposite these angles are equal:

• $AB = BP$.

Calculating Side Lengths

P is the midpoint of BC. This means:

• $BC = 2 \times BP$.

We know $BC = 10 \text{ cm}$.

• $10 \text{ cm} = 2 \times BP$.
• Solving for BP, we get $BP = \frac{10 \text{ cm}}{2} = 5 \text{ cm}$.

Since $AB = BP$:

• $AB = 5 \text{ cm}$.

Finally, since opposite sides of a parallelogram are equal ($CD = AB$):

• $CD = 5 \text{ cm}$.

Conclusion

The length of side CD is 5 cm.