In the given figure, 𝐴𝐵, 𝐶𝐷 and 𝐸𝐹 are parallel lines. Given 𝐴𝐵 = 7.5 𝑐𝑚, 𝐷𝐶 = 𝑦 𝑐𝑚, 𝐸𝐹 = 4.5 𝑐𝑚, 𝐵𝐶 = 𝑥 𝑐𝑚 and 𝐶𝐸 = 3 𝑐𝑚. Calculate the value of 𝑥 and 𝑦.
(Given: ∆FCD ~ ∆FAB, ∆BCD ~ BEF)
(Given: ∆FCD ~ ∆FAB, ∆BCD ~ BEF)
.Given, 𝐴𝐵, 𝐶𝐷 and 𝐸𝐹 are parallel lines. Given 𝐴𝐵 = 7.5 𝑐𝑚, 𝐷𝐶 = 𝑦 𝑐𝑚, 𝐸𝐹 = 4.5 𝑐𝑚, 𝐵𝐶 = 𝑥 𝑐𝑚 and 𝐶𝐸 = 3 𝑐𝑚.
To Find: Value of x and y.
We know that a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then the other two sides are divided in the same ratio.
Now, consider ΔFAB.
CD || AB [Given]
By BPT, we get
⇒ DF/BD = CF/AC...................................................... (i)
Since, ΔFCD ~ ΔFAB,
DF/BF = CD/AB = y/7.5
BF = 7.5 DF/y ............................................................... (ii)
Consider ΔBEF.
CD || EF [Given]
By BPT, we get
⇒ BD/DF = BC/CE = x/3 ............................................ (iii)
Since, ΔBCD ~ ΔBEF,
BD/BF = BC/BE = CD/EF
BD/BF = x/(x+3) = y/4.5 .............................................. (iv)
BF = 4.5 BD/y ................................................................(v)
From (ii) and (v),
7.5 DF/y = 4.5 BD/y
BD/DF = 7.5/4.5 = 5/3 ................................................... (vi)
From (iii) and (vi),
x/3 = 5/3
x = 5 cm
From (iv), we get
x/(x+3) = y/4.5
Substituing the value of x , we get
5/(5+3) = y/4.5
y = (5×4.5)/8 = 2.8125 cm
Hence, x = 5 and y = 2.8125.
To Find: Value of x and y.
We know that a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then the other two sides are divided in the same ratio.
Now, consider ΔFAB.
CD || AB [Given]
By BPT, we get
⇒ DF/BD = CF/AC...................................................... (i)
Since, ΔFCD ~ ΔFAB,
DF/BF = CD/AB = y/7.5
BF = 7.5 DF/y ............................................................... (ii)
Consider ΔBEF.
CD || EF [Given]
By BPT, we get
⇒ BD/DF = BC/CE = x/3 ............................................ (iii)
Since, ΔBCD ~ ΔBEF,
BD/BF = BC/BE = CD/EF
BD/BF = x/(x+3) = y/4.5 .............................................. (iv)
BF = 4.5 BD/y ................................................................(v)
From (ii) and (v),
7.5 DF/y = 4.5 BD/y
BD/DF = 7.5/4.5 = 5/3 ................................................... (vi)
From (iii) and (vi),
x/3 = 5/3
x = 5 cm
From (iv), we get
x/(x+3) = y/4.5
Substituing the value of x , we get
5/(5+3) = y/4.5
y = (5×4.5)/8 = 2.8125 cm
Hence, x = 5 and y = 2.8125.
