Rounded to the nearest tenth, what is the area of rectangle ABCD?
rectangle ABCD with diagonal AC, labeled as measuring 9 feet, angle BAC measuring 30 degrees, angle BCA measuring 60 degrees, angle DCA measuring 30 degrees, and angle DAC measuring 60 degrees
To find the area of the rectangle, we can first find the length and width of the rectangle using trigonometry.
Let's start by finding the length of the rectangle.
In triangle ABC, since angle BAC is 30 degrees and angle BCA is 60 degrees, the remaining angle A is 90 degrees.
Using the sine function, we can write:
sin(30 degrees) = opposite / hypotenuse
sin(30 degrees) = BC / 9
0.5 = BC / 9
BC = 4.5 feet
Similarly, in triangle CDA, since angle DCA is 30 degrees and angle DAC is 60 degrees, the remaining angle D is 90 degrees.
Using the sine function, we can write:
sin(60 degrees) = opposite / hypotenuse
sin(60 degrees) = DC / 9
√3/2 = DC / 9
DC = (9√3/2) feet ≈ 7.794 feet (rounded to the nearest thousandth)
The length of the rectangle is BC = 4.5 feet and the width of the rectangle is DC ≈ 7.794 feet.
Therefore, the area of the rectangle ABCD is approximately 4.5 feet * 7.794 feet = 35.073 feet² (rounded to the nearest tenth).