Final answer:
The shorter diagonal of a parallelogram can be calculated using the law of cosines on half of the diagonal formed by bisecting the parallelogram. Trigonometric principles allow us to determine this length precisely.
Step-by-step explanation:
To find the length of the shorter diagonal of a parallelogram with sides of lengths 10.3 cm and 23.2 cm and one of the angles being 54.2°, you can employ trigonometric principles and the law of cosines. The law of cosines states that for any triangle with sides a, b, and c, and the corresponding opposite angles A, B, and C, the following is true: c² = a² + b² - 2ab×cos(C).
In the context of a parallelogram, the diagonals bisect each other, creating two congruent triangles within the parallelogram. Thus, if one diagonal is d, the sides 10.3 cm and 23.2 cm would be a and b, and the included angle would be C (54.2°).
Using the law of cosines, we calculate the shorter diagonal (d), which is half of the actual diagonal of the parallelogram:
d² = 10.3² + 23.2² - 2×10.3×23.2×cos(54.2°).
After computing, we can take the square root to find the length of d. It's imperative to select the appropriate option that corresponds with the calculated value from the given choices. Remember, this calculated length is half the diagonal of the parallelogram, so the actual length of the diagonal will be twice the value of d.