Math Problem Statement

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

Solution

The image shows a parallelogram DEFGDEFG with diagonals that intersect at point PP. The dimensions and some angles within the figure are provided:

  • EF=4xEF = 4x inches
  • ED=4yED = 4y inches
  • DG=x+9DG = x + 9 inches
  • FG=3x+7FG = 3x + 7 inches
  • Angle PFG=113\angle PFG = 113^\circ

The problem asks to solve for DPDP, the distance from point DD to point PP, and it provides three answer choices: PGPG, FPFP, or FDFD.

Step 1: Recognizing the Parallelogram Properties

In parallelograms, the diagonals bisect each other. Therefore, the diagonal segments on either side of point PP must be equal. This gives us the relationships:

  • DP=PGDP = PG
  • EP=PFEP = PF

Step 2: Matching Segment Lengths

Given that the diagonals bisect each other, DP=PGDP = PG. Therefore, the correct value for DPDP is PGPG.

Thus, the answer is DP=PGDP = PG.


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 DEFGDEFG?
  4. How could we use the angles given in the problem to solve for xx and yy?
  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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Geometry
Parallelogram Properties
Diagonals Bisecting Theorem

Formulas

DP = PG (as diagonals of a parallelogram bisect each other)

Theorems

Diagonals Bisect Each Other in Parallelograms

Suitable Grade Level

Grades 9-10