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