If D is the midpoint of the hypotenuse AC of a right ABC, prove that BD=12AC.


Answer:


Step by Step Explanation:
  1. Let us plot the right ABC such that D is the midpoint of AC.
      A B C D
  2. We need to prove that BD=12AC.

    Let us draw a dotted line from D to E such that BD=DE and a dotted line from E to C.
      A B C D E


    In ADB and CDE, we have AD=CD[Given]ADB=CDE[Vertically opposite angles]BD=ED[By construction]ADBCDE[By SAS-criterion]
  3. As the corresponding parts of congruent triangles are equal, we have AB=CE and BAD=ECD

    Also, BAD and ECD are alternate interior angles. CEAB
  4. Now, CEAB and BC is a transversal. ABC+BCE=180[Co-interior angles]90+BCE=180[As ABC is right-angled triangle]BCE=90
  5. Now, in ABC and ECB, we have BC=CB[Common]AB=EC[By step 3]CBA=BCE [Each equal to 90]ABCECB[By SAS-criterion] As the corresponding parts of congruent triangles are equal, we have AC=EB12AC=12EBBD=12AC
  6. Thus, BD=12AC

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