`P` is the mid-point of side `B C` of a parallelogram `A B C D` such that `/_B A P=/_D A Pdot` If `A D=10 c m ,\ ` then `C D=`

Options

A. 5cm
B. 10cm
C. 6cm
D. 8cm

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- We have a parallelogram ABCD.
- P is the midpoint of side BC.
- The angles \( \angle BAP \) and \( \angle DAP \) are equal.
- The length of side AD is given as 10 cm.

2. **Understanding the Properties of Parallelogram:**
- In a parallelogram, opposite sides are equal and parallel.
- Therefore, \( AD \parallel BC \) and \( AB \parallel CD \).

3. **Using the Midpoint:**
- Since P is the midpoint of BC, we can say:
\[ BP = \frac{1}{2} BC \]
- Let's denote \( BC \) as \( x \). Then:
\[ BP = \frac{x}{2} \]

4. **Using the Angle Equality:**
- Since \( \angle BAP = \angle DAP \), we can conclude that triangle APB is isosceles because it has two angles equal.
- Therefore, the sides opposite these angles are equal:
\[ AB = AP \]

5. **Relating Sides Using the Isosceles Triangle Property:**
- In triangle APB, since \( BP = \frac{x}{2} \) and \( AB = AP \), we can say:
\[ AB = BP = \frac{x}{2} \]

6. **Using the Property of Parallelograms:**
- In parallelogram ABCD, opposite sides are equal, so:
\[ AB = CD \]
- Thus, we can write:
\[ CD = \frac{x}{2} \]

7. **Relating BC and AD:**
- Since \( AD \) is equal to \( BC \) in a parallelogram, we have:
\[ AD = x \]
- Given that \( AD = 10 \, \text{cm} \), we can substitute:
\[ x = 10 \, \text{cm} \]

8. **Finding CD:**
- Now substituting \( x \) back into the equation for \( CD \):
\[ CD = \frac{x}{2} = \frac{10}{2} = 5 \, \text{cm} \]

### Final Answer:
Thus, the length of \( CD \) is \( 5 \, \text{cm} \).