Answer:
9 units
Step-by-step explanation:
By the midsegment theorem,
[tex]AB=\dfrac{KN+LM}{2}[/tex]
Since the ratio [tex]LM:KN=1:5,[/tex] then [tex]LM=x,\ KN=5x[/tex] and
[tex]18=\dfrac{x+5x}{2}\\ \\18=\dfrac{6x}{2}\\ \\18=3x\\ \\x=6\\ \\LM=6\ units\\ \\KN=30\ units[/tex]
Draw two heights LE and MF. Quadrilateral ELMF is a rectangle, then EF = LM = 6 units.
Trapezoid KLMN is isosceles trapezoid (KL = MN), then KE = FN and
[tex]KE=FN=\dfrac{KN-EF}{2}=\dfrac{30-6}{2}=12\ units[/tex]
Consider right triangle KLE. By the Pythagorean theorem,
[tex]KL^2=LE^2+KE^2\\ \\15^2=LE^2+12^2\\ \\LE^2=15^2-12^2\\ \\LE^2=225-144=81\\ \\LE=9\ units[/tex]
