If the altitude from one vertex of a triangle bisects the opposite side, prove that the triangle is isosceles.


Answer:


Step by Step Explanation:
  1. We know that an altitude from a vertex to the opposite side is the perpendicular drawn from that vertex to the opposite side.

    Let us now draw the altitude AD from the vertex A of ABC to the opposite side BC.
      A B C D
    As the altitude AD bisects the opposite side BC, BD=DC.
  2. We need to prove that the triangle is isosceles, i.e., AB=AC.
  3. In ADB and ADC, we have BD=DC [Given] AD=AD [Common] ADB=ADC=90[As ADBC]ADBADC [By SAS congruence criterion] 
  4. As the corresponding parts of congruent triangles are equal, we have AB=AC.
    Thus, ABC is an isosceles triangle.

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