Final answer:
The question pertains to finding the lengths of the sides of a right triangle with angles measuring 30, 60, and 90 degrees, and with one side given as 63. Upon calculation, the side opposite the 30° angle is found to be 31.5, and the side opposite the 60° angle is 31.5√3. The choices provided by the student do not correspond to these calculated lengths.
Step-by-step explanation:
The student's question involves solving for components of a right triangle DEF with given measurements. Since angle E is 90 degrees, angle D must be 30 degrees because the angles in a triangle sum up to 180 degrees, and we already have a 60-degree angle at F and a 90-degree angle at E. Using the properties of a 30-60-90 right triangle, we can determine the length of the other sides if DE is the hypotenuse.
In a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2. If DE, the hypotenuse, measures 63, then the side opposite the 30-degree angle (DF) is half the hypotenuse, which is 31.5, and the side opposite the 60-degree angle (EF) is the hypotenuse times √3 / 2, which is 31.5√3. Thus, the correct answer is:
- DF (opposite the 30° angle) = DE/2 = 63/2 = 31.5
- EF (opposite the 60° angle) = DE√3/2 = (63√3)/2 = 31.5√3
Among the provided choices, b) 42√3 is incorrect, and c) 21√3 is incorrect. The choices seem to be incorrectly related to the question, as none of them matches the calculations we've done based on the given information.