Question

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.​

Answer 1

$\huge{\fbox{\fbox{\bigstar{\mathfrak{\red{Answer}}}}}}$

$According \:to$

$Thales\: theorem$

∆CDC ∼ ∆ABE

➼ $\frac{cd}{ab} = \frac{de}{be}$

➼ $\frac{y}{7.5} = \frac{de}{be}$

➼ $1 - \frac{y}{7.5} = 1 - \frac{de}{be}$

➼ $\frac{7.5 - y}{7.5} = \frac{be - de}{be}$

➼ $\frac{7.5 - y}{7.5} = \frac{bd}{be} ...... \: (1)$

Similarly ∆BDC ∼ ∆BEF

➼ $\frac{bd}{be} = \frac{cd}{ef} = \frac{bc}{bc + cf}$

➼ $\frac{bd}{be} = \frac{y}{4.5} = \frac{x}{x + 3} \: ....... \: (2)$

$From ( 1 ) and ( 2 )$

➼ $\frac{7.5 - y}{7.5} = \frac{y}{4.5}$

➼ $7.5y=(7.5)(4.5)-4.5y$

➼ $12y=(7.5)(4.5)=33.75$

➼ $y = \frac{33.75}{12} = 2.8125$

$From ( 2 )$

➼ $\frac{y}{4.5} = \frac{x}{x + 3}$

➼ $\frac{2.8}{4.5} = \frac{x}{x + 3}$

➼ $4.5 x = (2.8) x + 8.4$

➼$4.5 x - 2.8 x = 8.4$

➼ $1.7 x = 8.4$

➼ $x=\frac{8.4}{1.7}$

$\huge\pink{5\:(appr)}$

Answer 2

$\bf\large\underline\pink{Solution:-}$

Step 1: Look at the given figure and note down the given values

NOTE: AB || CD || EF

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

DC = y cm, and BC = x cm

Step 2: Identify the similar triangles in the given figure.

EXPLANATION:1) ∠ACB = ∠ECF (Vertical angles)

∠BAC = ∠FEC (Alternate angles)

∠ABC = ∠CFE (Alternate angles)

So, From AAA theorem Consider ΔACB and ΔCEF are similar,

2) From BPT theorem, consider Δ BCD and ΔBFE are similar triangles.

Step 3: Use the similar triangle properties

NOTE: the ratios of the lengths of similar triangles corresponding

sides are equal.

ΔACB and ΔCEF are similar,

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

EF

AB

=

3

x

Δ BCD and ΔBFE are similar

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

DC

BC

=

FE

BF

Step 4: Plugging the known values and simplify for unknown

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

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

4.5

7.5

=

3

x

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

4.5

7.5

∗3=x

x = 5

Step 5: Use this x value to find the y value

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

DC

BC

=

FE

BF

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

y

x

=

4.5

x+3

Substitute the x value

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

y

5

=

4.5

5+3

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

8

4.5

∗5

y = \frac{45}{16}y=

16

45