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AB = M₂M₁

Step-by-step explanation:

See attached diagram of trapezium ABCD.

(i) Triangle ABN and triangle CND are similar, so we can write a proportion:

AN / NC = AB / CD

AN / NC = a / 3a

AN / NC = 1 / 3

(ii) M₂ is the midpoint of AC, so:

2 AM₂ = AC

2 (AN + NM₂) = AN + NC

2 AN + 2NM₂ = AN + NC

AN + 2NM₂ = NC

1 + 2NM₂ / AN = NC / AN

1 + 2NM₂ / AN = 3

2NM₂ / AN = 2

NM₂ / AN = 1

Triangle ABN and triangle M₂M₁N are similar, so:

AB / M₂M₁ = AN / NM₂

AB / M₂M₁ = 1

AB = M₂M₁

the answer is 99.