Question

Right triangle ABC
has side lengths AB=6
, BC=63–√
, and AC=12
. A second right triangle, A′B′C′
, has side lengths of 83–√
, 8, and 16. Find the ratio of the side opposite ∠C
to the hypotenuse of triangle ABC
. Then use this ratio to identify the location of point C′
in the second right triangle.(1 point)
Responses

The ratio of the opposite side to the hypotenuse is 12
, and point C′
is opposite the side that has 83–√
.
The ratio of the opposite side to the hypotenuse is Start Fraction 1 over 2 End Fraction , and point upper C prime is opposite the side that has 8 Start Root 3 End Root .

The ratio of the opposite side to the hypotenuse is 3√2
, and point C′
is opposite the side that has 83–√
.
The ratio of the opposite side to the hypotenuse is Start Fraction Start Root 3 Emd Root over 2 End Fraction , and point upper C prime is opposite the side that has 8 Start Root 3 End Root .

The ratio of the opposite side to the hypotenuse is 12
, and point C′
is opposite the side that has length 8.
The ratio of the opposite side to the hypotenuse is Start Fraction 1 over 2 End Fraction , and point upper C prime is opposite the side that has length 8.

The ratio of the opposite side to the hypotenuse is 21
, and point C′
is opposite the side that has length 8.

Answer 1

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.**