Question

Determine the length AB in the figure below without using Cosine and Sine formulas?

Answer 1

To determine the length of AB without using trigonometry let us imagine C as the origin and the given line through C as X-axis .

Perpendiculars AM and BN are drawn on X-axis .
`angle DMC=angle DCM=pi/6` is drawn.

Now `DeltaAMD` becomes equilateral and `Delta DMC` is isosceles. So `AM=AD=DM=OD=1/2AB=500'`

Now `OM=sqrt(AO^2-AM^2)=500sqrt3`

So coordinates of `A->(-500sqrt3,500)`

Now in `Delta BON, angle BON=pi/4and angle BNO=pi/2`

So `ON=BN`

Hence `2BN^2=500^2`

`=>BN=500/sqrt2`

Hence coordinates of `B->(500/sqrt2,500/sqrt2)`

So length of AB

`=sqrt((500/sqrt2+500sqrt3)^2+(500-500/sqrt2)^2)`

`=500sqrt((1/2+3+sqrt6+1+1/2-sqrt2)`

`=500sqrt(5+sqrt6-sqrt2)~~1228.34'`