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

1 Answer
Feb 13, 2016

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)