The diagonals of a rhombus divide the rhombus into 4 congruent right triangles. Since angle A measures 120°, angle D also measures 120° and angles B and C each measure 60°. Each right triangle is made up of a right angle, a 30° angle (the angle formed by the diagonal bisecting the 60° angle of the rhombus), and a 60° angle (the angle formed by the diagonal bisecting the 120° angle of the rhombus). The bases of each triangle are made up of half the shorter diagonal (opposite the 30° angle) and half the longer diagonal (opposite the 60° angle). The hypotenuse is the side of the rhombus, 10 units. Since these right triangles are 30-60-90 right triangles, the measure of the shorter base is \(\frac{10}{2}\) = 5 units, and the measure of the longer base is 5√3 units. The longer base is half the length of the longer diagonal; therefore, the length of the longer diagonal is 2(5√3) = 10√3 units.

The cosine of ACD is equal to the length of \(\overline{CD}\) divided by the length of diagonal AC. Diagonal AC along with sides AD and DC form a right triangle. Use the Pythagorean theorem to find the length of \(\overline{AD}\): (\(\overline{AD}\))² + 12² = 13², (\(\overline{AD}\))² + 144 = 169, (\(\overline{AD}\))² = 25, \(\overline{AD}\)= 5 units. If the length of \(\overline{AD}\) is 5 units, then the length of BC is also 5 units. Since the length of \(\overline{CD}\) is 12 units, the length of AB is also 12 units. The perimeter of ABCD is 5 + 12 + 5 + 12 = 34 units. Notice that this is only one possible perimeter of ABCD. The cosine could have been given in reduced form. The length of \(\overline{CD}\) could be 24, and the length of \(\overline{AC}\) could be 26. However, since the length of every side of ABCD is an integer, the only possible perimeters are multiples of 34. Choice d is the only answer choice that is a multiple of 34.

The diagonal of a square is the hypotenuse of an isosceles right triangle, since the 4 angles of a square are right angles, and the sides of a square are all congruent to each other. Therefore, the measure of a side of the square is equal to the measure of the diagonal divided by √2:\(\frac{(2x-2)\sqrt{2}}{\sqrt{2}}\) = 2x – 2 units. Therefore, the perimeter of the square, 5x + 1, is equal to 2x – 2 + 2x – 2 + 2x – 2 + 2x – 2 = 8x – 8; 8x – 8 = 5x + 1, 3x = 9, x = 3. The length of a side of the square is equal to 2(3) – 2 = 6 – 2 = 4 units.

The tangent of angle ACB is equal to the length of \(\overline{AB}\) divided by the length of \(\overline{AB}\). Therefore, if the length of \(\overline{AB}\) is 8 units, then = 8, and \(\overline{AB}\)= 64. The perimeter of ABCD is equal to 64 + 8 + 64 + 8 = 144 units.