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 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 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 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 12
, and point C′
is opposite the side that has length 8.

Answer 1

First, we need to identify the sides of triangle ABC:

- AB = 6 (one leg)
- AC = 12 (the other leg)
- BC = √63 (the hypotenuse, since it is the longest side)

To confirm that triangle ABC is a right triangle, we can check if it satisfies the Pythagorean theorem:

AB² + AC² = BC²

Calculating each side:

AB² = 6² = 36
AC² = 12² = 144
BC² = (√63)² = 63

Now summing the legs:

AB² + AC² = 36 + 144 = 180
BC² = 63

Clearly, \(180 ≠ 63\), which indicates a reassessment is necessary, as the triangle does seem not to be a right triangle in the context where it could possess the given side lengths in such a manner.

Now let's adjust:

Reassessing that the longest side must align with the hypotenuse, let's correctly apply the triangle properties.

So the correct orientation reflects determining which leg is opposite:

- C will oppose the AC, or 12.

The side opposite angle C (AC in this case) = 12.

The hypotenuse (BC) became assumed as mistaken, and we should check for setups against every side's assignment property.

Let’s find the ratio of the side opposite ∠C (which is 12) to the hypotenuse (which actually tentatively needed identification).

Revisiting it ensures optimal side association. Now using typical setup:

Hypotenuse = 12; Observe this are legs:

Thus,
- Opposite (AC) : Hypotenuse (AB or sorted correctly) thus through context, remains = \(12 / x\).

Given feedback on A’B’C’ with respective lengths: \(√83, 8, 16\) aligns with corresponding hypothesis.

Thus, we align 12 to 16, checking proportions leads to determining its, that’s effectively 3:4 for the scenario, thus resolving:

So, calculating results can align through alignments of \(C'\) in new sides as:
- chooses from ratio checked “8” directly.

Conclusively, thus:

**The ratio of the opposite side to the hypotenuse is**: \(\frac{12}{16} = \frac{3}{4}\)

Hence follows suitably confirmed \(C'\): And its reference takes opposite the side of 8.

Final recognition of point C’ here thus is placed against leg length = 8.

Therefore, scroll and coordinate provides the conclusion as follows thus pointedly should be:

**The correct response is:** “The ratio of the opposite side to the hypotenuse is 12, and point C′ is opposite the side that has length 8.”