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,
![KL^2 =ML^2+MK^2,\\ \\15^2=(x+4)^2+MK^2.](/tpl/images/0503/0546/4a5db.png)
2. Consider right triangle KMN. By the Pythagorean theorem,
![MN^2 =NK^2+MK^2,\\ \\13^2=x^2+MK^2.](/tpl/images/0503/0546/e705e.png)
Subtract these two equations:
![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.](/tpl/images/0503/0546/a2dfe.png)
Then
![13^2=5^2+MK^2,\\ \\MK^2=169-25,\\ \\MK^2=144,\\ \\MK=12\ un.](/tpl/images/0503/0546/3341f.png)
The area of KLMN is
![A_{KLMN}=\dfrac{LM+KN}{2}\cdot MK=\dfrac{5+9}{2}\cdot 12=14\cdot 6=84\ un^2.](/tpl/images/0503/0546/a7ab6.png)
![Given: mk ⊥ lm , mk ⊥ kn mn=13, kl=15, lm-kn=4 find: area of klmn.](/tpl/images/0503/0546/9689c.jpg)