Prove that the sum of the lengths of the three altitudes of a triangle is less than the sum of the lengths of the three sides of the triangle.


Answer:


Step by Step Explanation:
  1. Let AP,AP, BQ,BQ, and CRCR be the altitudes of ABC.ABC.
      A B C P R Q
  2. We know that the perpendicular is the shortest line segment that can be drawn from a point outside the line to that line.

    Thus, we have AP<AB[ As AP is the perpendicular from the point A.]BQ<BC[ As BQ is the perpendicular from the point B.]CR<AC[ As CR is the perpendicular from the point C.] Adding the three equations, we have AP+BQ+CR<AB+BC+AC Sum of the lengths of the altitudes < Sum of the lengths of the sides 
  3. Thus, the sum of the lengths of the three altitudes of a triangle is less than the sum of the lengths of the three sides of the triangle.

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