Question

The question is below?

Let `ABC` and `ABC^'` be two non congruent triangles with sides `AB=4, AC=AC'=2sqrt2` and angle `/_B=30^@`. The absolute value of the difference between the areas of these triangles is?

Answer 1

The two non congruent triangles `DeltaABCandDeltaABC'` differ in sides BC and BC'.Their possible lengths can be found out by solving the following quadratic equation of `x` obtained by using cosine law of triangle. Here two roots of `x` represents the lengths of `BC andBC'`. Let `BC'>BC`

Now

`cos30^@=(x^2+4^2-(2sqrt2)^2)/(2*4*x)`

`=>sqrt3/2=(x^2+16-8)/(8x)`

`=>x^2-4sqrt3x+8=0`

`=>x=(4sqrt3pmsqrt((4sqrt3)^2-4*8*1))/2`

`=>x=2sqrt3pm2`

So `BC'=2sqrt3+2andBC=2sqrt3-2`

So `BC'-BC=4`

Now `DeltaABC'=1/2*AB*BC'*sin30^@`
`=1/2*4*BC'*1/2=BC'`

Similarly

`DeltaABC=1/2*AB*BC*sin30^@`
`=1/2*4*BC*1/2=BC`

So `DeltaABC'-DeltaABC=BC'-BC=4` squnit