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

Answer 1

To find the ratio of the side opposite ∠C to the hypotenuse of triangle ABC, we identify:

- The side lengths:
- AB = 6 (adjacent to ∠C)
- BC = \( 63 - \sqrt{63} \) (hypotenuse)
- AC = 12 (opposite to ∠C)

The side opposite ∠C is AC, which is 12. The hypotenuse is BC, which is \( 63 - \sqrt{63} \).

The ratio of the side opposite ∠C (AC) to the hypotenuse (BC) can be calculated as:

\[
\text{Ratio} = \frac{\text{AC}}{\text{BC}} = \frac{12}{63 - \sqrt{63}}
\]

Next, we check the second right triangle A′B′C′ with side lengths of \( 8, 16, \) and \( \sqrt{83} \).

We need to identify where point C′ is located opposite to based on the calculated ratio in triangle ABC.

1. The triangle's side lengths of A′B′C′ are:
- \( AB' = 8 \) (adjacent)
- \( AC' = 16 \) (hypotenuse)
- \( BC' = \sqrt{83} \) (opposite to angle C′)

2. For identifying point C′, we need to check the calculated ratio \( \frac{12}{63 - \sqrt{63}} \) in relation to the sides provided in triangle A′B′C′.

Given that:

\[
\frac{12}{63 - \sqrt{63}} \text{ simplifies to some specific fraction, and aligning it with aspects of triangle A′B′C′, C′ must align with side length 8.}
\]

3. Given the nature of the original question, the simplified ratio \( 12 \) would be of size \( 12:16 \) or \( \frac{12}{16} = \frac{3}{4} \).

Thus, confirming in the available responses, point C′ is opposite the side that has length 8.

With the given options:

\[
\text{The ratio of the opposite side to the hypotenuse is } 12, \text{ and point } C′ \text{ is opposite the side that has length } 8.
\]