Answer:
[tex]NO=10\ units\\ \\OP=18\ units[/tex]
Step-by-step explanation:
If [tex]NO:OP=5:9,[/tex] then
[tex]NO=5x\\ \\OP=9x[/tex]
Consider triangles LOP and LKM. In these triangles,
[tex]\angle L\cong \angle L[/tex] - reflexive property;
[tex]\angle LOP\cong \angle LKM[/tex] as corresponding angles when two parallel lines OP and KM are intersected by transversal KL;
[tex]\angle LPO\cong \angle LMK[/tex] as corresponding angles when two parallel lines OP and KM are intersected by transversal LM.
Hence, [tex]\triangle LOP\sim \triangle LKM[/tex] by AA similarity theorem. Similar triangles have proportional corresponding parts, so
[tex]\dfrac{OP}{KM}=\dfrac{\text{Height }\triangle LOP}{\text{Height }\triangle LKM}\\ \\ \\\dfrac{9x}{48}=\dfrac{16-5x}{16}\\ \\9x\cdot 16=48\cdot (16-5x)\\ \\144x=768-240x\\ \\144x+240x=768\\ \\384x=768\\ \\x=2\\ \\NO=5x=5\cdot 2=10\ units\\ \\OP=9\cdot 2=18\ units[/tex]
