Question

How to construct triangle LMN , such that LN = 8 cm and ∠ LMN = 80 degrees and LM- MN= 3 cm?

Answer 1

Use `LM=7.461cm` and `MN=4.461cm`

Explanation:

Calling LM as `x` and MN as `y`
`x-y=3` => `y=x-3`

As `/_ LMN` is the angle between x and y
Applying the Law of Cosines:

`8^2=x^2+y^2-2x*y*cos 80^@`
`64=x^2+(x-3)^2-2x(x-3)cos 80^@`
`x^2+x^2-6x+9-2x^2cos80^@+6xcos 80^@-64=0`
`(2-2cos 80^@)x^2+(-6+6cos 80^@)x-55=0`

`Delta=36+36cos^2 80^@-72cos 80^@+440-440cos80^@`
`Delta=476+36cos^2 80^@-512cos80^@`
`Delta~=388.178` => `sqrt(Delta)~=19.702`

`x=(6(1-cos 80^@)+-19.702)/(4(1-cos 80^@)`

`-> x_1=7.461cm` => `y=7.461-3` => `y=4.461cm`
`-> x_2=-4.461` (not valid)