# ABCD is a trapezium whose BC parallel AD and AD = 4 cm. Diagonals AC and BD are intersected at O in such way that #[AO]/[OC] = [DO]/[OB] = 1/2#, then what is the length of BC?

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Let ( x ) be the length of BC. Since BC is parallel to AD and ABCD is a trapezium, we know that AD and BC are the non-parallel sides. Let E be the point of intersection of AC and BD.

Given that [AO]/[OC] = 1/2, we can say that AO is 1/3 of AC and OC is 2/3 of AC. Similarly, since [DO]/[OB] = 1/2, we can say that DO is 1/3 of BD and OB is 2/3 of BD.

Since AC and BD are the diagonals of the trapezium, they bisect each other. Therefore, AE = EC and DO = OB.

Let's denote the length of AD as 4 cm. Using the above information, we can express the lengths of AE, EO, DO, and OB in terms of x:

- AE = EC = 4 cm (since AD = 4 cm)
- EO = OC = 2/3 * AC = 2/3 * (AD + DC) = 2/3 * (4 + x) = 8/3 + 2x/3 cm
- DO = OB = 1/3 * BD = 1/3 * (BC + CD) = 1/3 * (x + 4) = x/3 + 4/3 cm

Since EO and DO are equal, we have:

[ 8/3 + 2x/3 = x/3 + 4/3 ]

[ 6 + 2x = x + 4 ]

[ 2x - x = 4 - 6 ]

[ x = 2 ]

Therefore, the length of BC is 2 cm.

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Given that ( \frac{{AO}}{{OC}} = \frac{1}{2} ) and ( \frac{{DO}}{{OB}} = \frac{1}{2} ), and ( AD = 4 ) cm, we can use the properties of similar triangles to find the lengths of the sides of trapezium ( ABCD ).

Since ( \frac{{AO}}{{OC}} = \frac{1}{2} ), we can infer that triangles ( ABO ) and ( COO ) are similar. Similarly, since ( \frac{{DO}}{{OB}} = \frac{1}{2} ), triangles ( DOB ) and ( BOA ) are similar.

Let ( OC = x ). Then, ( AO = 2x ) and ( BO = 4x ) (since ( OB = 2AO )). Since ( AD = 4 ) cm, and ( AD = OC + AO ), we have ( 4 = x + 2x = 3x ), which implies ( x = \frac{4}{3} ) cm.

Now, we can find ( BO = 4x = \frac{16}{3} ) cm. And since ( BO = BC + CO ), we have ( BC = BO - OC = \frac{16}{3} - \frac{4}{3} = \frac{12}{3} = 4 ) cm.

Therefore, the length of ( BC ) is ( 4 ) cm.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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