The diagonals of rectangle ABCD intersect at point O. Find the perimeter of triangle AOB if CAD = 30 degrees, AC = 12cm.

Perimeter of triangle AOB = AB + BO + OA. Let’s find each side.
Consider a triangle ADC. It is rectangular, angle CAD = 30, hypotenuse AC = 12. Let’s find the side of CD: since the side opposite to the angle of 30 degrees is equal to half of the hypotenuse, then CD = 12: 2 = 6. Opposite sides in a rectangle are equal, so AB = CD = 6.
AO = OB = 6, since the diagonals are halved by the intersection point (12: 2 = 6).
Perimeter of triangle AOB = 6 + 6 + 6 = 18