Question

In the adjoining figure, AB || EF || CD, CD = 12 cm, AB = 7.2 cm and DF = 4.8 cm. Prove that `(CF)/(FB) = (DF)/(FA)`. Also, find the value of y, if x = 4.5 cm.

Answer 1

1. Prove the ratio `(CF)/(FB) = (DF)/(FA)`

Consider triangles ΔFAB and ΔFCD.

Since AB || CD, the alternate interior angles are equal:

∠FDC = ∠FBA

∠FCD = ∠FAB

Therefore, ΔFAB ∼ ΔFDC by the AA similarity criterion.

From this similarity, the ratios of their corresponding sides are equal:

`(CF)/(FB) = (CD)/(AB) = (DF)/(FA)`

Thus, we have proven:

`(CF)/(FB) = (DF)/(FA)`

2. Relate sides and find y

From the similarity ΔFAB ∼ ΔFDC,  we can substitute the known values CD = 12 cm and AB = 7.2 cm:

`(CF)/(FB) = 12/7.2`

Simplifying the ratio:

`(CF)/(FB) = 120/72 = 5/3`

This means CF = 5k and FB = 3k for some constant k. 

Therefore, the total length CB = CF + FB = 8k.

In ΔCBA, since EF || AB, ΔCEF ∼ ΔCBA.

Their sides are proportional:

`(EF)/(AB) = (CF)/(CB)`

Given EF = x = 4.5 cm and AB = 7.2 cm:

`4.5/7.2 = (5k)/(8k) = 5/8`

Verification: `4.5/7.2 = 45/72 = 5/8`, which is consistent.

Now, we apply the same logic to the other transversal BD.

In ΔBDC, since EF || CD, ΔBEF ∼ ΔBDC:

`(BF)/(BD) = (EF)/(CD)`

We know BF = y and FD = 4.8 cm,

So, BD = BF + FD

= y + 4.8

`y/(y + 4.8) = 4.5/12`

3. Calculate the value of y

Solve the equation for y:

12y = 4.5(y + 4.8)

12y = 4.5y + 21.6

7.5y = 21.6

`y = 21.6/7.5`

y = 2.88 cm

The value of y is 2.88 cm.