Answer:
84 square units.
Step-by-step explanation:
If LM - KN = 4, then denote KN = x and LM = x+4.
1. Consider right triangle KLM. By the Pythagorean theorem,
[tex]KL^2 =ML^2+MK^2,\\ \\15^2=(x+4)^2+MK^2.[/tex]
2. Consider right triangle KMN. By the Pythagorean theorem,
[tex]MN^2 =NK^2+MK^2,\\ \\13^2=x^2+MK^2.[/tex]
Subtract these two equations:
[tex]15^2-13^2=(x+4)^2-x^2,\\ \\225-169=x^2+8x+16-x^2,\\ \\56=8x+16,\\ \\8x=56-16,\\ \\8x=40,\\ \\x=5\ un.[/tex]
Then
[tex]13^2=5^2+MK^2,\\ \\MK^2=169-25,\\ \\MK^2=144,\\ \\MK=12\ un.[/tex]
The area of KLMN is
[tex]A_{KLMN}=\dfrac{LM+KN}{2}\cdot MK=\dfrac{5+9}{2}\cdot 12=14\cdot 6=84\ un^2.[/tex]
