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9514 1404 393

  BE = 266 2/3

  PE = 233 1/3

Step-by-step explanation:

Part A: The figure has 3 triangles, all similar. In order of shortest-to-longest side lengths, they are ...

  ΔBPE ~ ΔBCG ~ ΔGRE

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Part B: They can be declared similar by the AA postulate. For BPE and BCG, the angles at B are vertical angles, so are congruent. The angles at C and G are alternate interior angles, so are congruent.

For triangles BPE and GRE, angle E is congruent to itself, The angles at R and P are corresponding angles, so are congruent.

(Remember, GRPC is a parallelogram, so any line in the figure is a transversal crossing parallel lines.)

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Part C:

From parts A and B, we know ΔBPE ~ ΔBCG. Corresponding sides are proportional:

  BP/BC = BE/BG = PE/CG

  200/300 = BE/400 = PE/350

  BE = (400)(2/3) = 266 2/3

  PE = (350)(2/3) = 233 1/3

the answer will be the last one