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The triangles LMN and LPQ are illustrations of similar triangles
The length of line segment PM is 5.5 cm
The given parameters are:
LP = 2 cm
LQ = 3 cm
QN = 2 cm
m ∠LPQ = m ∠LNM
m ∠LQP = ∠LMN
The above parameters mean that, triangles LMN and LPQ are similar by the AA similarity theorem.
So, we have the following equivalent ratio
[tex]LP : LQ = LN : LM[/tex]
The ratio becomes
[tex]2 :3 = 5 : LM[/tex]
The segment LM is the sum of LP and PM.
So, we have:
[tex]2 :3 = 5 : LP + PM[/tex]
Express as fraction
[tex]\frac{2}{3} = \frac{5}{ LP + PM}[/tex]
Substitute 2 for LP
[tex]\frac{2}{3} = \frac{5}{2 + PM}[/tex]
Cross multiply
[tex]4 + 2PM = 15[/tex]
Subtract 4 from both sides
[tex]2PM = 11[/tex]
Divide both sides by 2
[tex]PM = 5.5[/tex]
Hence, the length of line segment PM is 5.5 cm
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