SOLUTION: please help me solve this complicated math problem: Let ABCD be a parallelogram, with M the midpoint of DA, and diagonal AC of length 36. Let G be the intersection of MB and AC. Wh

Algebra ->  Parallelograms -> SOLUTION: please help me solve this complicated math problem: Let ABCD be a parallelogram, with M the midpoint of DA, and diagonal AC of length 36. Let G be the intersection of MB and AC. Wh      Log On


   

Question 1028303: please help me solve this complicated math problem: Let ABCD be a parallelogram, with M the midpoint of DA, and diagonal AC of length 36. Let G be the intersection of MB and AC. What is the length of AG?
Answer by ikleyn(52800) About Me  (Show Source):
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please help me solve this complicated math problem: Let ABCD be a parallelogram, with M the midpoint of DA,
and diagonal AC of length 36. Let G be the intersection of MB and AC. What is the length of AG?
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Make a sketch and label the points.

Notice that the triangles AGM and CGB are similar.

   Indeed, they have congruent corresponding angles:

      - angles AGM and CGB are vertical angles;
      - angles MAG and GCB are alternate interior angles at two parallel lines AD and BC and the transverse AC;
      - angles AMG and GBC are alternate interior angles at two parallel lines AD and BC and the transverse BM.

Therefore, the corresponding sides of the triangles AGM and CGB are proportional.

In particular, since AM is half of AD, then AG is half of CG.

Hence, AC is three times as long as AG: |AC| = 3*|AG|.

Then |AG| = abs%28AC%29%2F3 = 36%2F3 = 12 units long.

Solved.