Krishna
0

Step 1: Explore the given question and note down the given data

            NOTE: ABC is a triangle right angled at B, and D and E are points

                        of trisection of BC.

            GIVEN: 8AE^2 = 3 AC^2 + 5 AD^2   

            ASSUME: Let BD = DE = EC = x

                            Then BE = 2x and BC = 3x


Step 2: Apply the Pythagoras theorem to the right angle triangles in the given figure .

        EXAMPLE:  In Δ ABD,

                               AD^2 = AB^2 + BD^2

                               AD^2 = AB^2 + x^2

                            In Δ ABE,

                               AE^2 = AB^2 + BE^2

                               AE^2 = AB^2 + (2x)^2

                               AE^2 = AB^2 + 4x^2 ...........................(1)

                            In Δ ABC,

                       AC^2 = AB^2 + BC^2 

                       AC^2 = AB^2 + (3x)^2

                       AC^2 = AB^2 + 9x^2


Step 3: Substitute the known values in the given equation and simplify.

            GIVEN: 8AE^2 = 3 AC^2 + 5 AD^2   

                                       = 3(AB^2 + 9x^2) + 5(AB^2 + x^2)

                                      = 3 AB^2 + 27x^2 + 5AB^2 + 5 x^2

                                      =      8 AB^2 + 32 x^2

                                      = 8(AB^2 + 4x^2             

                                       =   8 AE^2     {since equation (1)}     

                                                  Hence proved..