1, because point d is the midpoint of AC and DC=2cm.

So AC=2DC=4cm.

Because BC= 1/2AB, AC=AB+BC.

So AC=3BC=4cm.

So BC=4/3cm.

That is AB=2BC=8/3cm.

2. Because 3AC=2AB

So AC=2/3AB.

Because CB=AB-AC=AB-2/3AB= 1/3AB.

DC = AC-AD = AC- 1/2AB = 2/3AB- 1/2AB = 1/6AB

DE = DC+CE = DC+ 1/2CB = 1/6AB+ 1/2 * 1/3AB = 1/3AB = 6

So AB= 18

Because AD= 1/2AB=9.

CB= 1/3AB=6

So AD:CB=9:6=3:2.

3. Because ∠α=2∠β, three times the complementary angle of ∠αis equal to the complementary angle of ∠β.

So (90-∠ α) * 3 = 180-∠β.

(90 -2∠β)*3= 180 -∠β

5∠β=90

∠β= 18

∠α=2* 18 =36

4. Because of OE⊥CD

So ∠ doe = ∠ DOB+∠ BOE = 90.

Because OF⊥ AB

So ∠ BOF = ∠ BOD+∠ DOF = 90.

So ∠ BOE = ∠ BOF = 65.

(2) Because AB pays the CD at O point.

So ∠AOC=∠BOD

Because OF⊥ AB

So ∠ BOF = ∠ BOD+∠ DOF = 90.

Because ∠ DOF is 65.

So ∠ BOD = 90-65 = 25.

That is ∠ AOC = 25.

5、∠BAC=∠ACB=60

BC=30 nautical miles

You can see this by drawing a picture.

6. I don't know, sorry