Answer a geometry question

In a triangle ABC,the point Y divides the side AB in the ratio 2:1, The point Z on AC is such that YZ is parallel to BC.
Give 2 options that are the same as the ratio BC : YZ.
with workings please

∆ABC and ∆AYZ are similar, so
BC : YZ = 3 : 2

notice that 1.5 : 1 = 3:2 , so it looks like choices 1.5:1 and 3:2 are the required ratios

1) Consider the two triangles, ABC & AYZ:

i) ii) iii)
==> Both triangles are similar to each other [AAA similarity axiom]

2) If two triangles are similar, then their corresponding sides are in proportion.

Since triangle ABC is similar to triangle AYZ,

AB/AY = BC/YZ = CA/ZA ---- (i)

But, given AY/YB = 2/1; ==> BY/YA = 1/2

==> (BY+YA)/AY = (1+2)/2

==> AB/AY = 3/2 ---- (ii)

Hence from (i) & (ii) above,

AB/AY = BC/YZ = CA/ZA = 3/2

There are two triangle
1. ABC
2.AYZ

Which are possing geometrical symmetry(there is one theorem which says that if two triangle have same angles{at least 2} then they called geo.. Sym... And here it happens)

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so we can take respective side ratio:

BC:YZ=AB:AY=AC:AZ.