Final answer:
In the given rectangle, the length of side BC can be calculated using the sine rule in triangle ACD to be approximately 13.86 cm. The length of diagonal AC, which can be calculated using Pythagoras' theorem in triangle ABC, is approximately 18.48 cm.
Step-by-step explanation:
In the rectangle ABCD, AB = 12 cm represents the length of the rectangle, and angle BAC = 30° is the angle between diagonal AC and side AB. To calculate the lengths of side BC and diagonal AC, we can use the properties of the rectangle along with trigonometry.
Firstly, because ABCD is a rectangle, AB = DC and AD = BC. Since AB = 12 cm, DC = 12 cm. Also, because angle BAC = 30° (which implies angle CAD = 60° because the angles in a triangle sum up to 180°), we can use the sine rule in triangle ACD to find the length of side AD, which also equals the length of side BC. The sine rule states that the ratio of a side of a triangle to the sine of the opposite angle is the same for all 3 sides. So, we get AD/BC = AB/sin(CAD), i.e., BC = 12/sin(60) = 13.86 cm approximately.
Now, we calculate the length of the diagonal, AC, using Pythagoras' theorem (since triangle ABC is a right-angle triangle). So, we get AC = sqrt(AB^2 + BC^2) = sqrt((12)^2 + (13.86)^2) = 18.48 cm approximately.
Thus, the length of side BC is 13.86 cm and that of diagonal AC is 18.48 cm.
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