https://file.aiquickdraw.com/mathbot/file/1728561242158fkhxb245.jpg

The image shows a parallelogram \( DEFG \) with diagonals that intersect at point \( P \). The dimensions and some angles within the figure are provided:

- \( EF = 4x \) inches
- \( ED = 4y \) inches
- \( DG = x + 9 \) inches
- \( FG = 3x + 7 \) inches
- Angle \( \angle PFG = 113^\circ \)

The problem asks to solve for \( DP \), the distance from point \( D \) to point \( P \), and it provides three answer choices: \( PG \), \( FP \), or \( FD \).

### Step 1: Recognizing the Parallelogram Properties
In parallelograms, the diagonals bisect each other. Therefore, the diagonal segments on either side of point \( P \) must be equal. This gives us the relationships:
- \( DP = PG \)
- \( EP = PF \)

### Step 2: Matching Segment Lengths
Given that the diagonals bisect each other, \( DP = PG \). Therefore, the correct value for \( DP \) is \( PG \).

Thus, the answer is **\( DP = PG \)**.

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Would you like to go into more detail about how we used the properties of parallelograms to solve this?

Here are 5 related questions you can explore:
1. How can the bisecting property of diagonals in parallelograms help in solving other geometry problems?
2. What other properties are unique to parallelograms, and how are they useful?
3. How would you calculate the perimeter of parallelogram \( DEFG \)?
4. How could we use the angles given in the problem to solve for \( x \) and \( y \)?
5. How can you verify that diagonals are equal in length using algebraic methods?

**Tip**: In parallelograms, diagonals always bisect each other, meaning the two segments of each diagonal are equal in length. This is a useful property when solving for unknown lengths.

Refer to parallelogram DEFG. What is the value of DP?

In this geometry problem, learn how to solve for the length of DP in parallelogram DEFG using the bisecting property of diagonals. Explore step-by-step how the diagonals divide each other into equal segments. Suitable for Grades 9-10.

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