We can check each option.
a) We have to prove that x = 8.
The sides KN and MN should have the same length, so we can write:
[tex]\begin{gathered} KN=MN \\ 7x-4=5x+12 \\ 7x-5x=12+4 \\ 2x=16 \\ x=\frac{16}{2} \\ x=8 \end{gathered}[/tex]As x has to be 8, this option is true.
b) This option states that KN = 24.
We can test this using the value of x we have just calculated:
[tex]\begin{gathered} KN=7x-4 \\ KN=7\cdot8-4 \\ KN=56-4 \\ KN=52 \end{gathered}[/tex]This option is false, as KN = 52.
c) As MN = KN, and KN = 52 (from the last calculation), we can conclude that MN = 52.
Then, this option is true.
d) This option states that KL ≅ ML.
By symmetry, the two sides have to have the same length and are congruent, so this option is true.
e) This option states that LM ≅ LN.
This two segments can not be equal, as they are part of a right angle and the hypotenuse as LN can not have the same length as a leg, like LM.
This option is false.
Answer: the true options are A, C and D.