Question

Four sides of a hexagon are given as 5 cm, 3 cm, 6 cm and 7 cm in order. If angles of the hexagon are congruent, find the remaining sides.​

Answer 1

Given :  Four sides of a hexagon are given as 5 cm, 3 cm, 6 cm and 7 cm in order  . angles of the hexagon are congruent,

To find : remaining sides.​

Solution:

angles of the hexagon are congruent,

Hence 120° Each

Let say A =  (0 , 0)

AB = 5 cm  

B =  ( 5 , 0)

BC = 3 cm

C  = ( 5 + 3Cos60   , 0 + 3Sin60 )  

=> C = ( 13/2  ,  3√3 / 2)

CD  = 6 cm

D = 13/2 - 6/2   ,  3√3 / 2 + 6*√3 / 2    

=> D =  7/2  ,  9√3 / 2

DE = 7 cm

=> E =  7/2 - 7  ,  9√3 / 2

=> E = ( -7/2 , 9√3 / 2  )

Let say  F = ( x , y)  

A = (0 , 0)

Slope AF  = -√3

=>  y/x =   -√3    =>  y = - √3  x

Slope of EF  = √3

E = ( -7/2 , 9√3 / 2  )  

√3   =  ( y -  9√3 / 2 ) / (x - ( -7/2))

=> √3  ( x +  7/2)    = ( y - 9√3 / 2  )

Substitute y = - √3  x

=> √3  ( x +  7/2)    = (  - √3  x  - 9√3 / 2  )

=> x + 7/2 =  -x  -  9/2

=> 2x  = -8

=> x = - 4

=> y = 4√3

F  = ( - 4  , 4 √3 )

E = ( -7/2 , 9√3 / 2  

EF =   √(- 4 - (-7/2))² + ( 4 √3 - 9√3 / 2)²  = 1

AF =    √(-4)² +(4√3)² = 8

Length  of remaining sides

= 1 &  8

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