Krishna
0

Step 1: Recall the properties of similar figures

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


              Properties:

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


                 \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=\frac{22.5}{8}

                   y = 2.8125


           Hence,  x = 5 and y = 2.8