Krishna
0

Step 1: Recall the basic proportionality theorem

            If two triangles are similar then,

                  I) Corresponding angles are equal for both triangles

                  ii) The corresponding sides of the two triangles are in proportion to one another


            Basic proportionality theorem:

            If a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides

            are divided in the same ratio.


Step 1: Examine the given figure and note down the given data

                  

                  In the given figure, LM || CB and  LN || CD

                   Prove    \frac{AM}{MB} = \frac{AN}{AD}


Step 2: Divide the given figure into two triangles based up on the given data

              From the figure:                  

                      The triangle are ABC and triangle ADC

                      In triangle ABC LM || CB  (lines are parallel)

                      In triangle ABD LN || CD  (lines are parallel)


                      Hence, we can apply basic proportionality theorem.

                    

Step 3: Apply the basic proportionality theorem to the triangles.

                              In triangle ABC, 

                      If LM || CB,  then the other two sides are divided in the same ratio.

                                       \frac{AM}{MB} = \frac{AL}{LC}.......................(1)         \because Basic proportionality theorem


                                In triangle ADC,


                      if LN || CD then the other two sides are divided in the same ratio.

                                     \frac{AN}{ND} = \frac{AL}{LC} ........................(2)                       \because Basic proportionality theorem


Step 4:  Prove the given ratio by using the equations      

                      From the equations (1) and (2)

                               \frac{AM}{MB} = \frac{AL}{LC}.......................(1)


                                \frac{AN}{ND} = \frac{AL}{LC} ........................(2)    


                Substitute equation (1)( \frac{AL}{LC} value) in equation (2)


                      We can write   \frac{AM}{MB} = \frac{AN}{ND}          


                      Hence, proved   \frac{AM}{MB} = \frac{AN}{ND}