Question

calculate the value of x and y

Answer 1

Answer :

In this figure

AB  | |  CD  | |  EF

And

Given AB= 7.5 cm, DC= y cm, EF= 4.5 cm, BC=x cm and CE= 3 cm

Now in ∆ BFE , we know CD  | |  EF , So from basic proportionality theorem , we get

BD/DF = BC/CE⇒BD/DF = x3
So,

BD  =  x and DF  = 3

Now in ∆ FBA , we know CD  | |  AB , So from basic proportionality theorem , we get

FD/CD = FB/AB⇒FD/CD = (FD + DB)/AB⇒3/y = (x + 3)/7.5              −−−−−−−−− ( 1 )

And

Now in ∆ BFE , we know CD  | |  EF , So from basic proportionality theorem , we get

BC/CD = BE/EF⇒BC/CD = (BC + CE)/EF⇒x/y = (x + 3)/4.5           ⇒y = 4.5x/(x + 3)                 −−−−−−−−− ( 2 )

Now we substitute value from equation 2  , in equation 1 , we get

⇒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 = 0⇒4.5x2 − 9x  − 67.5 = 0⇒x2 −2x  − 15 = 0

From, Splitting the middle term method we get

⇒ x2 - 5x  + 3x  - 15  = 0

⇒x ( x  - 5 ) + 3 ( x - 5 ) = 0

⇒ ( x - 5 ) ( x + 3 ) = 0
So,
x  = 5  And  - 3 

But side of triangle can't be negative , So

x  = 5  , Substitute that value in equation 2 , we het

y  = 4.5 × 5/5 + 3 = 22.5/8 = 225/80 = 45/16 = 2.8125 
So,

x  = 5  cm
And
y  = 2.8125  cm                                                                       ( Ans )

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