To solve the problem, we first identify the properties of triangle ABC and find the ratio of the side opposite angle C to the hypotenuse.
Triangle ABC
- The sides are given as:
- \( AB = 6 \)
- \( AC = 12 \)
- \( BC = 63 - \sqrt{} \) (assuming you mean it to be a certain value, but let's assume it's computed properly)
Given that ABC is a right triangle, we can verify that it satisfies the Pythagorean theorem: \[ AB^2 + BC^2 = AC^2 \]
Calculating: \[ 6^2 + BC^2 = 12^2 \] \[ 36 + BC^2 = 144 \] \[ BC^2 = 144 - 36 = 108 \] \[ BC = \sqrt{108} = 6\sqrt{3} \]
Ratio Calculation for Triangle ABC
We now find the ratio of the side opposite angle C (which is AB = 6) to the hypotenuse (which is AC = 12): \[ \text{Ratio} = \frac{AB}{AC} = \frac{6}{12} = \frac{1}{2} \]
Identifying Point C' in Triangle A'B'C'
Triangle A'B'C':
- The sides are given as:
- \( A'B' = \sqrt{83} \) (opposite side),
- \( A'C' = 16 \) (hypotenuse),
- \( B'C' = 8 \).
Since we established that the known ratio from triangle ABC is \( \frac{1}{2} \), we can look for the side opposite \( C' \) which satisfies this ratio.
Knowing that \( A'C' = 16 \) is the hypotenuse, we compute the opposite side corresponding to this hypotenuse: \[ \text{Opposite side} = \frac{1}{2} \times 16 = 8 \]
Conclusion
The ratio of the side opposite angle C in triangle ABC to the hypotenuse is \(\frac{1}{2}\). Consequently, point \( C' \) is opposite the side that has length \( 8 \).
Thus, the correct response is: The ratio of the opposite side to the hypotenuse is \( \frac{1}{2} \), and point C′ is opposite the side that has length 8.