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

Answer:

Step-by-step explanation:

In ΔBEF, from basic proportionality theorem.  

BD/DF = BC/CE

BD/DF = x/3

or BD= x and DF= 3

In ΔFBA  

FD/CD = FB/AB

FD/CD= FD+DB/AB

3/y = x +3/7.5

In ΔBEF

BC/CD= BE/EF

BC/CD= BC=CE/EF

x/y = x+3/4.5

y= 4.5x/ x+3

By subsituting values

3/4.5x/x+3 = x+3/7.5

3x+9/ 4.5x = x+3/7.5

22.5x + 67.5= 4.5x² + 13.5x

4.5x² + 13.5x - 22.5x - 67.5

4.5x² - 9x - 67.5=0

x² -2x -15=0

By factorizing

x² -5x +3x -15=0

(x-5) (x+3)=0

x= 5 and -3

triangle can not be negative so,  

y= 4.5 x 5/ 5+3

y=2.8125

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