Draw the normal from vertex L to side MK and let it meet MK at P.
ΔLMP is 45-45-90 right triangle => LM(hypotenuse)= 12 . So legs MP=LP= 12/√2 = 6√2.
ΔKLP is 60-30-90 right triangle => larger leg= LP=6√2. smaller leg, KP= largerleg / √3 = 2√6.
LK = hypotenuse 2 x smaller leg = 4√6.
MK= MP+KP = 6√2 + 2√6
Perimeter = 12 + 4√6 + 6√2 + 2√6 = 12+6√6+6√2 = 6(2+√6+√2) = 35.18
2nd Part:
Draw NP perpendicular to MO, where P is a point on MO.
Using 45-45-90 special right triangle rule on triangle MNP, NP = MN/√2 = 3√2 =MP.
triangle ONP is 30-60-90 triangle. ON = 2 NP = 6√2, OP = NP√3= 3√6.
ON = 6√2 and MO =MP +OP =3√2 +3√6
