Question

In the following figure, AB || EF || CD. If AB = 7.5 cm, CD = 4.5 cm, EC = 3 cm, EF = x and BE = y, find x and y.

Answer 1

Since AB || EF || CD, the three parallel lines cut the transversals AD and BC proportionally.

So, `(AE)/(ED) = (BE)/(EC)`

Also, the lengths between the same two parallels are proportional:

AB : EF : CD

Hence,

`(AE)/(ED) = (AB)/(CD) = 7.5/4.5 = 5/3`

Therefore, `(BE)/(EC) = 5/3`

Given EC = 3 cm,

`BE = 5/3 xx 3`

BE = 5 cm

So, y = 5cm

Now to find EF, note that the middle parallel segment is the harmonic division between the two ends here, or more directly from similar triangles:

`(EF)/(AB) = (ED)/(AD) and (EF)/(CD) = (BE)/(BC)`

Using AE : ED = 5 : 3, we get

AD = AE + ED

`(ED)/(AD) = 3/(5 + 3)`

`(ED)/(AD) = 3/8`

Thus,

`EF = 3/8 xx AB`

`EF = 3/8 xx 7.5`

∴ EF = 2.8125

This does not match the figure’s intended standard proportional setup, so using the parallel-line ratio directly between the verticals:

Since AB, EF, and CD are parallel intercepts between the two transversals AD and BC, the middle intercept is proportional to the product over sum:

`EF = (AB xx CD)/(AB + CD)`

`EF = (7.5 xx 4.5)/(7.5 + 4.5)`

`EF = 33.75/12`

∴ EF = 2.8125 cm

So, x = 2.8125 cm, y = 5 cm or `x = 45/16  cm, y = 5  cm`.