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Question 15: In triangle MNL,line NP bisects the angle MNL. If NP : NL = 2 : 3 and angle MNL= 120 degrees . Then NP: NL: MN is:
We see that NP divides the angle MNL into two equal parts, so we use the angle bisector theorem.
So, MPPL = MNNL
In triangle NPL, PN : NL :: 2x : 3x
So lets say PN : NL : LP :: 2x : 3x : kx
Using cosine rule we can say that,
cos 60 = 12 = 22+32โk22ร2ร3
5
= 12 = 22+32โk22ร2ร3
6 = 13 โ k2
So, k = โ7
Lets say the side MN is kx,
So, from angle bisector theorem:
MPPL = MNNL
= MPโ7 = kx3x
MP = โ7รk3
Now in triangle MNP,
Using cosine rule:
Cos 60 = 12 = k2+22โ(โ7k/3)22ร2รk
We get
2 k = k2 + 4 - 79 k2
2 k = 29 k2 + 4
Solving this quadratic equation,
k2+ 18 = 9 k
(k-3)(k-6) = 0
So k could be 3 or 6
Since the triangle cannot be isosceles, k cannot be 3
So k = 6
This, PN : NL : LP :: 2x : 3x : 6x
Or PN : NL : LP :: 2 : 3 : 6
The question is "The ratio NP: NL: MN is:"
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