In the given figure, AB // CD // EF, given AB = 7.5 cm DC = y cm, EF = 4.5 cm, BC = x cm. Calculate the values of "x" and "y"

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