The triangles LMN and LPQ are illustrations of similar triangles
The length of line segment PM is 5.5 cm
How to determine the length of line segment PM?
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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