Answer:
[tex]JN \approx 6.86\;\; or\;\; JN = \dfrac{144}{21}[/tex]
Step-by-step explanation:
There are two different ways we could find segment JN. Since trapezoid LMPN is similar to LMKJ, we can create a proportion using the given equality statement to find segment JN.
Given proportion:
[tex]\displaystyle \frac{LJ}{JN} =\frac{MK}{KP}[/tex]
Substitute known values:
[tex]\displaystyle \frac{16-JN}{JN} =\frac{12}{9}[/tex]
Cross multiply:
[tex]\displaystyle JN * 12 = 9*(16-JN)[/tex]
[tex]\displaystyle 12JN = 144-9JN[/tex]
Add 9JN to both sides of the equation:
[tex]\displaystyle 21JN = 144[/tex]
Divide both sides of the equation by 21:
[tex]\displaystyle JN = 6.8571[/tex]
Again, since trapezoid LMPN is similar to LMKJ, we can create a proportion. However, this time we will use [tex]\frac{LN}{JN} =\frac{MP}{KP}[/tex], we know this is true since the figures are similar.
Proportion:
[tex]\displaystyle \frac{LN}{JN} =\frac{MP}{KP}[/tex]
Substitute known values:
[tex]\displaystyle \frac{16}{JN} =\frac{12+9}{9}[/tex]
Addition:
[tex]\displaystyle \frac{16}{JN} =\frac{21}{9}[/tex]
Cross multiply:
[tex]21 * JN = 16 * 9[/tex]
[tex]21JN = 144[/tex]
Divide both sides of the equation by 21:
[tex]\displaystyle JN = 6.8571[/tex]
You can see that both of these methods produce the same result, it depends upon which way you want to go about it.
