Final answer:
Without the length of the rectangular backyard, selecting the correct option for the measurement of the shorter sides is not possible based on the perimeter alone. However, none of the options are correct if the total perimeter is to be 32 1/2 meters and we are to use two equal shorter sides.
Step-by-step explanation:
To determine which option is correct for the measurement of the two shorter sides in the rectangle, we use the formula for the perimeter of a rectangle, which is P = 2l + 2w where P is the perimeter, l is the length, and w is the width.
For the rectangular backyard with a perimeter of 32 1/2 meters, if we let w represent the shorter sides, then we would rearrange the formula to solve for w as follows: w = (P - 2l) / 2. Without knowing the length, we cannot solve for w with the given options. However, by evaluating the options given, none of them would be correct if we strictly interpret the question as-presented.
For various related examples, when the dimensions are known, such as a rectangle with the length of 36.7 cm and width of 128.8 cm, the area can be calculated by multiplying these two dimensions. Alternatively, for a farmer measuring a triangular piece of land using displacement vectors and calculating the length and orientation of a third side, principles of vector addition and trigonometry can be applied.