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)ResponsesThe 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

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)ResponsesThe 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

GPT-4o mini · Answer

To find the ratio of the side opposite angle \( C \) to the hypotenuse of triangle \( ABC \), we need to identify which side corresponds to the angle \( C \) and what the lengths of the sides are.

In triangle \( ABC \):

\( AB = 6 \) (one leg)
\( AC = 12 \) (other leg)
\( BC = 63 - \sqrt{ } \) (hypotenuse)

Angle \( C \) is opposite side \( AB \), so the side opposite angle \( C \) is \( AB = 6 \). The hypotenuse is \( BC = 63 - \sqrt{ } \).

Now, we calculate the ratio:

\[ \text{Ratio} = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{6}{63 - \sqrt{ }}. \]

We need to simplify this ratio. Let's analyze the numerical values.

Before we determine the values further, we need to confirm the position of point \( C' \) in triangle \( A'B'C' \): Given triangle \( A'B'C' \) has sides of \( 8 \), \( 16 \), and \( \sqrt{83} \) (where \( 16 \) would be opposite to \( C' \)), we can analyze the ratios in that triangle.

Since the ratio found from triangle \( ABC \) doesn’t yield a clean numeric value due to having an \( \sqrt{ } \) term (which seemed to not be fully defined), we’d rather work with the angles and their properties in right triangles.

However, if we just simply write out that ratio again, focusing primarily on the lengths of triangle \( A'B'C' \) (which follow Pythagorean Triples), we stress the side opposite point \( C' \) which turns out to be \( 8 \) (the shorter leg).

That leads us to conclude: The ratio of the opposite side to the hypotenuse is essentially extracted albeit versus how we calculated with traditional definitions and adjustment to angles, the direction leads to:

The ratio of the opposite side to the hypotenuse is \( \left(\frac{3}{8}\right) \) aligning indirectly to give a relationship concerning which leg aligns correctly.

Thus, overall when trying to relate them, the response will be:

The ratio of the opposite side to the hypotenuse is \( \frac{6}{(63 - \sqrt{})} \), however in referencing toward location of point \( C' \): point \( C' \) is opposite the side that has length \( 8 \).