Final answer:
To find the length of the adjacent side of a parallelogram, we used the area formula to find the height when one side is the base. We then used trigonometry (sine function) to find the adjacent side length given the 54° angle and the height of 6 cm, resulting in approximately 7.4 cm.
Step-by-step explanation:
To find the length of one of the adjacent sides of a parallelogram with an area of 42 cm², a 54° angle, and one side that is 7 cm long, we utilize the formula for the area of a parallelogram, which is base × height.
The height here is the perpendicular distance from the base to the opposite side, which can be found using trigonometry.
The formula for the area can be rearranged to solve for the height: height = area ÷ base.
Thus, height = 42 cm² ÷ 7 cm = 6 cm.
This is the length of the side adjacent to the 54° angle when it acts as the height of the parallelogram.
Now we use the sine function to find the height: sin(54°) = perpendicular (height) ÷ hypotenuse (adjacent side). Rearranging this, we get the adjacent side length = height ÷ sin(54°).
With the given information, the adjacent side equals 6 cm ÷ sin(54°).
Calculating this we get approximately 7.4 cm to one decimal place.