Step 1: Recall the properties of similar figures

             Similar figures have the same shape but not necessarily the same size.


                1. Corresponding sides are proportional. (The ratios of the corresponding sides are equal.)

                2. Corresponding angles are congruent.

Step 2: Look at the given figure and note down the  given values


               From the figure:

                         AB || CD || EF

                            AB = 7.5 cm, CF = 3 cm, EF = 4.5 cm

                            DC = y cm, and BC =  x cm

Step 3: Identify the similar triangles in the given figure.

            From the figure:

                   \angle ACB = \angle ECF (Vertical angles)

                   \angle BAC = \angle FEC (Alternate angles)

                   \angle ABC = \angle CFE (Alternate angles)

               So, From AAA theorem, ΔACB and ΔCEF are similar

              CD \parallel EF   and  Δ BCD and ΔBFE 

                   \angle B = \angle B (common angle)                                   

                   \angle BCD = \angle BFE (Corresponding angles)

                 \angle BDC = \angle BEF   (Corresponding angles)

                According to the AAA theorem, Δ BCD and Δ BFE  are similar triangles.

Step 4: Use the similar triangle properties          

              ΔACB and ΔCEF are similar

                    So,  \frac{AB}{EF} = \frac{BC}{CF}                              \because     corresponding sides

              Δ BCD and ΔBFE  are similar

                    So, \frac{BC}{BF}=\frac{DC}{FE}                 \because   corresponding sides


Step 5: Plugging the known values and simplify for unknown

               We Known that  AB = 7.5 cm, EF = 4.5 cm

                       \frac{AB}{EF} = \frac{BC}{CF}     

                       \frac{AB}{EF} = \frac{x}{3}

                       \frac{7.5}{4.5} = \frac{x}{3}

                       \frac{7.5}{4.5} * 3 = x


                       x = 5

Step 6: Use this x value to find the y value      


                 \frac{BC}{DC} = \frac{BF}{FE}

                  \frac{x}{y} = \frac{x+3}{4.5}

                 Substitute the x = 5 value  

                  \frac{5}{y} = \frac{5+3}{4.5}

                   y = \frac{4.5}{8} *5


                   y = 2.8125

           Hence,  x = 5 and y = 2.8