To solve the problem based on the provided images and the information given about the parallelogram $ABCD$ and the diagonals intersecting at point $P$, we need to use the properties of diagonals in a parallelogram.

In any parallelogram, the diagonals bisect each other. This means: $AP = PD \quad \text{and} \quad BP = PC$

1. Length of $AD$:

Since the diagonals bisect each other, we know: $AB = AD$ From the figure: $AB = 19 \, \text{in}$ Thus, $AD = 19 \, \text{in}$

2. Length of $AP$:

We know that diagonals are bisected, meaning: $AP = PD$ Also, the total length of diagonal $AD$ is 19 inches. So: $AP = \frac{AD}{2} = \frac{19}{2} = 9.5 \, \text{in}$

3. Length of $CD$:

Since opposite sides of a parallelogram are equal, we have: $CD = AB = 19 \, \text{in}$

4. Length of $BD$:

We can find the length of diagonal $BD$ by adding up the given lengths of its segments: $BD = BP + PD = BP + AP$ From the figure, we have: $BP = 16 \, \text{in}, \quad AP = 9.5 \, \text{in}$ Thus: $BD = 16 + 9.5 = 25.5 \, \text{in}$

Final Answers:

• $AD = 19 \, \text{in}$
• $AP = 9.5 \, \text{in}$
• $CD = 19 \, \text{in}$
• $BD = 25.5 \, \text{in}$

Would you like further clarification or details on any of the steps? Here are some related questions to deepen your understanding:

1. How do diagonals of a parallelogram generally behave in terms of symmetry?
2. What other properties can be derived from the intersection of diagonals in a parallelogram?
3. How does the relationship between opposite sides in a parallelogram help in solving for other unknowns?
4. What methods would you use to calculate the area of this parallelogram?
5. Can we apply this method of bisecting diagonals to other quadrilaterals?

Tip: Always remember that the diagonals of a parallelogram bisect each other, which is a key property when solving such problems.