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 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 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 21
, and point C′
is opposite the side that has length 8.

Answer 1

To find the ratio of the side opposite angle \( C \) to the hypotenuse of triangle \( ABC \), we first need to identify the lengths of the sides corresponding to this angle.

In triangle \( ABC \):
- The sides are as follows:
- \( AB = 6 \)
- \( BC = 63 - \sqrt{} \)
- \( AC = 12 \)

### Step 1: Identify the hypotenuse
In a right triangle, the hypotenuse is the longest side. Among \( 6, 12, \) and \( 63 - \sqrt{} \), we must determine which side is the hypotenuse. Since \( \sqrt{} \) seems incomplete, we will ignore mistyping and consider that \( 12 \) is the largest known length.

Thus, the hypotenuse \( AC = 12 \).

### Step 2: Identify the opposite side
Here, angle \( C \) is opposite side \( AB \):
- Therefore, the side opposite \( C \) is \( AB = 6 \).

### Step 3: Calculate the ratio
The ratio of the side opposite angle \( C \) (which is \( AB \)) to the hypotenuse \( AC \) is:

\[
\text{Ratio} = \frac{AB}{AC} = \frac{6}{12} = \frac{1}{2}
\]

### Step 4: Applying the ratio to triangle \( A'B'C' \)
In triangle \( A'B'C' \), we have the sides:
- \( A'B' = 8 \)
- \( A'C' = 16 \)
- Let's denote \( A'B'' = \sqrt{83} \)

The hypotenuse is \( A'C' = 16 \).

To find where point \( C' \) is located, we see that \( C' \) must be opposite the side equivalent to the opposite side ratio from triangle \( ABC \). Since we have a ratio of \( \frac{1}{2} \), we can identify that the side opposite \( C' \) in triangle \( A'B'C' \):

\[
\frac{8}{16} = \frac{1}{2}
\]

Thus, point \( C' \) is opposite side \( 8 \).

### Final Answer:
The ratio of the opposite side to the hypotenuse is \( \frac{1}{2} \), and point \( C' \) is opposite the side that has length \( 8 \).