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

1 Answer
Rui D.
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)

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