1.

Let A, B, C, D, E represent vertices of a regular pentagon ABCDE. Given the position vector of these vertices be veca, veca + vecb, vecb, lamda veca andlamda vecb, respectively. AD divides EC in the ratio

Answer»

`cos""(2pi)/(5) : 1`
`cos""(3pi)/(5) : 1`
`1 : 2 cos""(2pi)/(5)`
`1 : 2`

Solution :Given ABCDE is a regular pentagon

Let position vector of POINT A and C be `VECA and veb`, respectively.
`AD` is parallel to BC and AB is parallel to EC.
Therefore,
AOCB is a PARALLELOGRAM
and position vector of B is `veca + VECB`
The position vectors of E and D are `lamdavecb and lamda veca`respectively.
Also `OA = BC = AB= OC =1` (let)
Therefore, AOCB is rhombus.
`angle ABC = angle AOC = (3pi)/(5)`
and `angleOAB = angle BCO = pi - (3pi)/(5) = (2pi)/(5)`
Further `OA = AE = 1 and OC = CD = 1 `
Thus, `DeltaEAO and Delta OCD` are isosceles.
In `Delta OCD`, using sine rule we get `(OC)/(sin""(2pi)/(5))= (OD )/( sin""(pi)/(5))`
`rArr OD = (1)/( 2 cos ""(pi)/(5)) = OE`
`rArr AD = OA + OD = 1 + (1)/(2 cos ""(pi)/(2))`
`rArr (AD)/(BC) =1 + (1)/(2 cos""(pi)/(5)) = (1 +2 cos""(pi)/(5))/( 2 cos""(pi)/(5))`
And `(OE)/( OC) = (1)/( 2cos ""(pi)/(5))`


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