P is any point in the interior of triangle ABC. Prove that AB+AC>BP+CP.


Answer:


Step by Step Explanation:
  1. Let us first mark the point P in the interior of ABC.
      A B C P
  2. Now, let us join line BP and CP and produce CP to meet AB at Q.
      A B C PQ
  3. We know that the sum of two sides of a triangle is greater than the third side.
    Thus in ACQ, we have AC+AQ>CQAC+AQ>CP+PQ(1) Similarly in BPQ, we have BQ+PQ>BP(2)
  4. By adding (1) and (2), we get:AC+AQ+BQ+PQ>CP+PQ+BPAC+AQ+BQ>CP+BP[Subtrating PQ from both the sides]AC+AB>CP+BP[As AQ + BQ = AB]
  5. Thus, we have AC+AB>CP+BP.

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