In △ A B C, let D be a point on BC such that AD bisects ∠ A. If A D = 6, B D = 4 and D C = 3, then find AB.

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In   △  A  B  C, let D be a point on BC such that AD bisects   ∠  A. If   A  D  =  6,   B  D  =  4 and   D  C  =  3, then find AB.

edgarovhg · Accepted Answer

Step 1Let E is point of line AD such that   ∠  A  E  B  =  90      °   and F is point of line AD such that   ∠  A  F  C  =  90      °  . Then triangles ABE and ACF are similar, triangles DBE and DCF are similar. Using this facts, one can write             A      F              A      E        =            C      F              B      E        =            C      D              B      D        =      3    4              D      F              D      E        =            C      D              B      D        =      3    4  Step 2Then             E      F              A      E        =  1  −            A      F              A      E        =      1    4              E      F              D      E        =  1  +            D      F              D      E        =      7    4              A      E              D      E        =            E      F              /            D      E              E      F              /            A      E        =  7            A      D              D      E        =            A      E              D      E        −  1  =  6  D  E  =  A  D      /    6  =  1  A  E  =  7  ⋅  D  E  =  7  B      E    2    =  B      D    2    −  D      E    2    =  16  −  1  =  15  A      B    2    =  A      E    2    +  B      E    2    =  49  +  15  =  64  A  B  =  8

equissupnica7 · Accepted Answer

Step 1With angle bisector theorem you can easily find out that             A      B              A      C        =      4    3  . By this relationship, you can label sides   A  B  =  4  x  ,  A  C  =  3  x.Draw perpendicular DK, DL such that points D, L lies on the sides AB, AC respectively.You can easily find out that   △  A  K  D  ≅  △  D  L  A (A. A. S.)By this, you can label   K  D  =  D  L  =  h and   A  K  =  A  L  =  aStep 2By the Pythagorean theorem, you can get the following equations,      h    2    +      a    2    =  36 (1)      h    2    +  (  3  x  −  a      )    2    =  9 (2)      h    2    +  (  4  x  −  a      )    2    =  16 (3)Expanding (2), (3) and substituting (1) as required will lead you to   x  =  2As   A  B  =  4  x  ,  A  B  =  2  ∗  4  =  8

In triangle ABC, let D be a point on BC such that AD bisects angle A. If AD=6, BD=4 and DC=3, then find AB.

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